Abstract
We establish a Rademacher type theorem involving Hamiltonians H(x, p) under very weak conditions in both of Euclidean and Carnot-Carathéodory spaces. In particular, H(x, p) is assumed to be only measurable in the variable x, and to be quasiconvex and lower-semicontinuous in the variable p. Without the lower-semicontinuity in the variable p, we provide a counter example showing the failure of such a Rademacher type theorem. Moreover, by applying such a Rademacher type theorem we build up an existence result of absolute minimizers for the corresponding \(L^\infty \)-functional. These improve or extend several known results in the literature.
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1 Introduction
Let \( \Omega \subset {{{\mathbb R}}^n}\) be a domain (that is, an open and connected subset) of \({{{\mathbb R}}^n}\) with \(n\ge 2\). We first recall Rademacher’s theorem in Euclidean spaces. See Appendix for some of its consequence related to Sobolev and Lipschitz spaces.
Theorem 1.1
If \(u:{\Omega }\rightarrow {\mathbb R}\) is a Lipschitz function, that is,
then, at almost all \(x\in \Omega \), u is differentiable and \(|\nabla u(x)| ={\mathrm {\,Lip}}u(x)\). Here \(|\nabla u(x)|\) is the Euclidean length of the derivative \(\nabla u(x)\) at x, and \({\mathrm {\,Lip}}u (x)\) is the pointwise Lipschitz constant at x defined by
The above Rademacher’s theorem was extended to Carnot-Carathéodory spaces \((\Omega , X)\), where X is a family of smooth vector fields in \({\Omega }\) satisfying the Hörmander condition (See Sect. 2). Denote by Xu the distributional horizontal derivative of \(u\in L^1_{\mathrm {\,loc\,}}(\Omega )\). Write \({d_{CC}}\) as the Carnot-Carathéodory distance with respect to X. One then has the following; see [19, 21, 24, 36] and, for the better result in Carnot group and Carnot type vector field, see [40, 42].
Theorem 1.2
If \(u:\Omega \rightarrow {\mathbb R}\) is a Lipschitz function with respect to \({d_{CC}}\), that is,
then, \(Xu \in L^\infty (\Omega ,{\mathbb R}^m)\) and, for almost all \(x\in \Omega \), the length \(|X u(x)| \le \lambda \).
Under the additional assumption that X is a Carnot type vector field in \(\Omega \), or in particular, \((\Omega , X)\) is a domain in some Carnot group, one further has \(|X u(x)| = {\mathrm {\,Lip}}_{d_{CC}}u(x) \) for almost all \(x\in \Omega \), where \({\mathrm {\,Lip}}_{d_{CC}}u(x) \) is defined by (1.2) with \(|y-x|\) replaced by \(d_{CC}(y,x)\).
This paper aims to build up some Rademacher type theorem involving Hamiltonians H(x, p) in both of Euclidean and Carnot-Carathéodory spaces. Throughout this paper, the following assumptions are always held for H(x, p).
Assumption 1
Suppose that \(H:\Omega \times {\mathbb R}^m\rightarrow {[0,+\infty )}\) is measurable and satisfies
-
(H1)
For each \(x\in \Omega \), \(H(x,\cdot )\) is quasi-convex, that is,
$$\begin{aligned} H(x,t p + (1 - t)q) \le \max \{H(x,p), H(x,q)\}, \quad \forall p, q \in {\mathbb R}^m, \ \forall \, t\in [0,1] \ \text {and} \ \forall x\in \Omega . \end{aligned}$$ -
(H2)
For each \( x\in \Omega \), \(H(x,0)=\min _{p\in {{{\mathbb R}}^m}}H(x,p)=0\).
-
(H3)
It holds that \(R_\lambda < \infty \) for all \(\lambda \ge 0\), and \(\lim _{\lambda \rightarrow \infty }R_\lambda ' = \infty \), where and in below,
$$\begin{aligned} R_\lambda := \sup \{|p| \ | \ (x,p)\in \Omega \times {\mathbb R}^{m},H(x,p)\le \lambda \} \end{aligned}$$and
$$\begin{aligned} R'_\lambda := \inf \{|p| \ | \ (x,p)\in \Omega \times {\mathbb R}^{m},H(x,p)\ge \lambda \}. \end{aligned}$$
For any \(\lambda \ge 0\), we define
Recall that \(\dot{W}^{1,\infty }_{X }(\Omega )\) denotes the set of all functions \(u\in L^\infty (\Omega )\) whose distributional horizontal derivatives \(Xu\in L^\infty (\Omega ;{\mathbb R}^m)\). It was known that any function \( u\in \dot{W}^{1,\infty }_{X }(\Omega )\) admits a continuous representative \(\widetilde{u}\); see [21, Theorem 1.4] and also Theorem 2.3 and Remark 2.4 below. In this paper, in particular, in (1.4) above, we always identify functions in \(\dot{W}^{1,\infty }_X(\Omega )\) as their continuous representatives. We remark that \(d_\lambda \) is not a distance necessarily, but Lemma 2.9 says that \(d_\lambda \) is always a pseudo-distance as defined in Definition 2.8 below.
Given any \(\lambda \ge 0\), by the definition (1.4), if \(u\in \dot{W}_{X}^{1,\infty }(U)\) and \(\Vert H(\cdot ,Xu)\Vert _{L^\infty (U)}\le \lambda \), then \( u(y)-u(x)\le d_{\lambda }(x,y) \ \forall \, x,y\in \Omega .\) It is natural to ask whether the converse is true or not. However, such converse is not necessarily true as witted by the Hamiltonian
for details see Remark 1.9 below. The point is that \( \lfloor |p|\rfloor \) is not lower-semicontinuous. Below, the converse is shown to be true if H(x, p) is assumed additionally to be lower-semicontinuous in the variable p, that is,
-
(H0)
For almost all \(x\in \Omega \), \(H(x,p)\le \liminf _{q\rightarrow p}H(x,q) \quad \forall p\in {\mathbb R}^m.\)
Theorem 1.3
Suppose that H satisfies (H0)-(H3). Given any \(\lambda \ge 0 \) and any function \(u:\Omega \rightarrow {\mathbb R}\), the following are equivalent:
-
(i)
\(u\in \dot{W}_{X}^{1,\infty }(\Omega ) \ \text {and} \ \Vert H(\cdot ,Xu)\Vert _{L^\infty (\Omega )}\le \lambda \);
-
(ii)
\(u(y)-u(x)\le d_\lambda (x,y)\) \(\forall x,y\in \Omega \);
-
(iii)
For any \(x\in \Omega \), there exists a neighborhood \(N(x)\subset \Omega \) such that
$$\begin{aligned} u(y)-u(z)\le d_\lambda (z,y)\quad \forall y,z \in N(x). \end{aligned}$$
In particular, if \(u:\Omega \rightarrow {\mathbb R}\) satisfies any one of (i)-(iii), then
Using Theorem 1.3, when \(\lambda \ge \lambda _H\) we prove that \(d_\lambda \) has a pseudo-length property, which allows us to get the following. Here and below, define
Since \(R_\lambda '\) defined in (H3) of Assumption 1 is always nonnegative and increasing in \(\lambda \ge 0\) and tends to \(\infty \) as \(\lambda \rightarrow \infty \), we know that \(0\le \lambda _H<\infty \), and moreover, \(\lambda >\lambda _H\) if and only if \(R'_\lambda >0\).
Theorem 1.4
Suppose that H satisfies (H0)-(H3). Given any \(\lambda \ge \lambda _H\) and any function \(u:\Omega \rightarrow {\mathbb R}\), the statement (i) in Theorem 1.3 is equivalent to the following
-
(iv)
For any \(x\in \Omega \), there exists a neighborhood \(N(x)\subset \Omega \) such that
$$\begin{aligned} u(y)-u(x)\le d_\lambda (x,y)\quad \forall y \in N(x). \end{aligned}$$
In particular, if \(u:\Omega \rightarrow {\mathbb R}\) satisfies (iv), then
As a consequence of Theorem 1.3 and Theorem 1.4, we have the following Corollary 1.5. Associated to Hamiltonian H(x, p), we introduce some notion and notations. Denote by \(\dot{W}^{1,\infty }_H(\Omega )\) the collection of all \(u\in \dot{W}^{1,\infty }_X(\Omega )\) with \(\Vert H(\cdot ,Xu)\Vert _{L^\infty (\Omega )}<\infty \). Denote by \({\mathrm {\,Lip}}_H(\Omega ,X)\) the class of functions \(u:\Omega \rightarrow \infty \) satisfying (ii) for some \(\lambda >0\) equipped with (semi-)norms
Denote by \({\mathrm {\,Lip}}_H^*( \Omega )\) the collection of all functions u with
where we write the pointwise “Lipschitz" constant
Thanks to the right continuity of the map \(\lambda \in [\lambda _H,\infty )\mapsto d_\lambda (x,y)\) as given in Lemma 4.3, the infima in (1.9) and (1.10) are actually minima.
Corollary 1.5
Suppose that H satisfies (H0)-(H3) with \(\lambda _H=0\). Then \(\dot{W}^{1,\infty }_H(\Omega )={\mathrm {\,Lip}}_H(\Omega )={\mathrm {\,Lip}}_H^*( \Omega )\) and
Next, we apply the above Rademacher type property to study a minimization problem for \(L^\infty \)-functionals corresponding to the above Hamiltonian H(x, p) in both Euclidean and Caratheódory spaces:
Aronsson [1,2,3, 5] in 1960’s initiated the study in this direction via introducing absolute minimizers. A function \(u\in W^{1,\infty }_{X,{\mathrm {\,loc\,}}}(U)\) is called an absolute minimizer in U for H and X (write \(u\in AM(U;H,X)\) for short) if for any domain \(V\Subset U\), it holds that
Here and throughout this paper, for domains A and B, the notation \(A\Subset B\) stands for that A is a bounded subdomain of B and its closure \(\overline{A}\subset B\).
The existence of absolute minimizers with a given boundary value has been extensively studied. Apart from the pioneering work by Aronsson mentioned above, we refer the readers to [7, 9, 12, 14, 15, 26, 33] and the references therein in the Euclidean setting. For the existence results in Heisenberg groups, Carrot-Carathéodory spaces and general metric spaces with special type of Hamiltonians, we refer the readers to [10, 34, 36,37,38, 47]. Usually, there are two major approaches to obtain the existence of absolute minimizers. When dealing with \(C^2\) Hamiltonians, one usually transfers the study of absolute minimizers into the study of viscosity solutions of the Aronsson equation (the Euler-Lagrange equation of the \(L^\infty \)-functional \({\mathcal F}\)). Thus, to get the existence of absolute minimizers, it suffices to show the existence of the corresponding viscosity solutions. This approach was employed, for instance, in [3, 4, 9, 15, 25, 33, 47]. To study the the existence of absolute minimizers for Hamiltonians H(x, p) with less regularity, one efficient way is to use Perron’s method to first get the existence of absolute minimizing Lipschitz extensions (ALME), and then show the equivalence between ALMEs and absolute minimizers. This idea was adopted in [1, 2, 7, 26, 34, 37, 38]. To see the close connection between ALMEs and absolute minimizers, we refer the readers to [17, 35, 36] and references therein.
Theorem 1.3, Theorem 1.4 and Corollary 1.5 allow us to apply Perron’s method directly and then to establish the following existence result of absolute minimizers. This is partially motivated by [12]. However, since we are faced with measurable Hamiltonians, there are several new barriers to be overcome as illustrated at the end of this section.
Theorem 1.6
Suppose that H satisfies (H0)-(H3) with \(\lambda _H=0\). Given any domain \(U\Subset {\Omega }\) and \(g\in {\mathrm {\,Lip}}_{d_{CC}}(\partial U)\), there must be a function \(u\in AM(U;H,X)\cap {\mathrm {\,Lip}}_{d_{CC}}(\overline{U})\) so that \(u|_{\partial U}=g\).
Theorem 1.3, Theorem 1.4, Corollary 1.5 and Theorem 1.6 improve or extend several previous studies in the literature including Theorem 1.1 and Theorem 1.2 above; see Remark 1.7 and Remark 1.8 below.
Remark 1.7
-
(i)
In Euclidean spaces, that is, \(X=\{\frac{\partial }{\partial x_i} \}_{1\le i\le n}\), if \(H(x,p)=|p|\), then Corollary 1.5 coincides with Lemma 6, which is a consequence of Theorem 1.1. In Carnot-Carathéodory spaces \((\Omega , X)\), if \(H(x,p)=|p|\), then Corollary 1.5 coincides with Lemma 2.7, which is a consequence of Theorem 1.2.
-
(ii)
In Euclidean spaces, if H(x, p) is lower semi-continuous in \(U \times {{{\mathbb R}}^n}\), \(H(x,\cdot )\) is quasi-convex for each \(x \in U\), and satisfies (H2) and (H3), (ii)\(\Leftrightarrow \)(i) in Theorem 1.3 was proved in Champion-De Pascale [14] (see also [6, 15] for convex H(p) in Euclidean spaces).
The proof in [14] relies on the lower semi-continuity in both of x and p heavily, which allows for approximation H(x, p) via a continuous Hamiltonian in x and p. But such an approach fails under the weaker assumptions (H0) &(H1) here. We refer to Sect. 7 for more details and related further discussions.
-
(iii)
In both Euclidean and Carrot-Carathéodory spaces, if \(H(x,p)=\sqrt{\langle A(x)p,p\rangle }\), where A(x) is a measurable symmetric matrix-valued function satisfying uniform ellipticity, then Theorem 1.3 was established in [36, 38]. The proofs therein rely on the inner product structure, and also do not work here. For measure spaces endowed with strongly regular nonlocal Dirichlet energy forms, where the Hamiltonian is given by the square root of Dirichlet form, we refer to [22, 37, 38, 44] for some corresponding Rademacher type property.
-
(iv)
Under merely (H0)-(H3), one can not expect that \({\mathrm {\,Lip}}_H(x)=H(x,Xu(x)) \) almost everywhere. Recall that in Euclidean spaces, there does exist A(x), which satisfying Remark 1.7(iii) above, so that such pointwise property fails for the Hamiltonian \(\sqrt{\langle A(x)p,p\rangle }\). For more details see [36, 46].
Remark 1.8
-
(i)
In Euclidean spaces, that is, \(X=\{\frac{\partial }{\partial x_i}\}_{1\le i\le n}\), if H(x, p) is given by Euclidean norm and also any Banach norm, the existence of absolute minimizers was established in Aronsson [1, 2] and Aronsson et al [7]. If H(x, p) is given by \(\sqrt{\langle A(x)p,p\rangle }\) with A being as in Remark 1.7 (iii) above, existence of absolute minimizers is given by [36] with the aid of [35]. In a similar way, with the aid of [35], Guo et al [26] also obtained the existence of absolute minimizers if H(x, p) is a measurable function in \(\Omega \times {{{\mathbb R}}^n}\), and satisfies that \(\frac{1}{C}< H(x,p)<C\) for all \(x\in \Omega \) and \(p\in S^{n-1}\), where \(C\ge 1\) is a constant, and that \(H(x,\eta p) = |\eta | H(x,p)\) for all \(x\in \Omega \), \(p\in {{{\mathbb R}}^n}\) and \(\eta \in {\mathbb R}\).
-
(ii)
In Euclidean spaces, if H(x, p) is continuous in both variables x, p and quasi-convex in the variable p, with an additional growth assumption in p, Barron et al [9] built up the existence of absolute minimizers. If H(x, p) is lower semi-continuous in x, p and quasi-convex in p and satisfies (H1)-(H3), Champion-De Pascale [14] established the existence of absolute minimizers with the help of their Rademacher type theorem in Remark 1.7(ii). Recall that the lower semi-continuity of H plays a key role in [14] to obtain the pseudo-length property for \(d_\lambda \).
-
(iii)
In Heisenberg groups, if \(H(x,p)=\frac{1}{2}|p|^2\) we refer to [10] for the existence of absolute minimizers. In any Carnot group, if \(H\in C^2(\Omega \times {\mathbb R}^m)\), \(D^2_{pp}H(x, \cdot )\) is positive definite, and there exists \( \alpha \ge 1\) such that
$$\begin{aligned} H(x,\eta p) = \eta ^{\alpha }H(x,p)\quad \forall x\in \Omega ,\ \eta >0,\ p\in {{{\mathbb R}}^n}, \end{aligned}$$(1.11)then the existence of absolute minimizers was obtained by Wang [47] via considering viscosity solutions to the corresponding Aronsson equations.
The following remark explains that, without the assumption (H0), Theorem 1.3 does not necessarily hold.
Remark 1.9
The Hamiltonian \(H(x,p)=\lfloor |p|\rfloor \) given in (1.5) satisfies (H1)-(H3) but does not satisfy (H0). Given any \(\lambda \in (0,1)\), we have
Fix any \( z\in \Omega \) and write \(u(x)=d_\lambda (z,x)\ \forall x\in \Omega \). By the triangle inequality we have
Recall that, when \(X=\{\frac{\partial }{\partial x_j}\}_{1\le j\le n}\) or when X is given by Carnot type Hörmander vector fields, one always has \(|X d_{CC} (z,\cdot ) | =1\) almost everywhere; see [40]. For such X, we conclude that
Thus Theorem 1.3 fails.
The following remark explains the reasons why we need \(\lambda \ge \lambda _H \) in Theorem 1.4, and why we assume \(\lambda _H=0\) in Theorem 1.6. Note that, in Theorem 1.3 where \(\lambda _H\) maybe not 0, we do get the equivalence among (i), (ii) and (iii) for any \(\lambda \ge 0\).
Remark 1.10
-
(i)
To prove (iv) in Theorem 1.4\(\Rightarrow \) (i) in Theorem 1.3, we need a pseudo-length property for \(d_\lambda \) as in Proposition 4.1. When \(\lambda >\lambda _H\) (equivalently, \(R'_\lambda >0\)), to get such pseudo-length property for \(d_\lambda \), our proof does use \(R_\lambda '>0\) so to guarantee that the topology induced by \(\{d_\lambda (x ,\cdot )\}_{x \in \Omega }\) (See Definition 2.8) is the same as the Euclidean topology; see Remark 4.4. When \(\lambda =\lambda _H\), we get such pseudo-length property for \(d_{\lambda _H}\) via approximating by \(d_{\lambda _H+\epsilon }\) with sufficiently small \(\epsilon >0\).
-
(ii)
When \( \lambda _H>0\) and \(0\le \lambda <\lambda _H\), we do not know whether \(d_\lambda \) enjoys such pseudo-length property. We remark that there does exist some Hamiltonian H(x, p) which satisfies the assumptions (H0)-(H3) with \(\lambda _H>0\) (that is, for some \(\lambda >0\), \(R_\lambda '=0\)) ; but for \(0<\lambda <\lambda _H\), the topology induced by \(\{d_\lambda (x ,\cdot )\}_{x \in \Omega }\) does not coincide with the Euclidean topology; see Remark 2.11 (ii).
-
(iii)
To get the existence of absolute minimizers, our approach does need Theorem 1.4 and also several properties of \(d^U_\lambda \), whose proof relies heavily on the pseudo-length property for \(d_\lambda \) and \(R_\lambda '>0\). In Theorem 1.6, we assume \(\lambda _H=0\) so that we can work with all Lipschitz boundary g so to get existence of absolute minimizer.
In the case \(\lambda _H>0\), our approach will give the existence of of absolute minimizer when the boundary \(g:\partial U\rightarrow {\mathbb R}\) satisfies \(\mu (g,\partial U)>\lambda _H\), but does not work when \(\mu (g,\partial U)\le \lambda _H\). Here \( \mu (g,\partial U)\) is the infimum of \(\lambda \) so that \(g(y)-d(x)\le d^U_\lambda (x,y)\) \(forall x,y\in \partial U\).
The paper is organized as below, where we also clarify the ideas and main novelties to prove Theorem 1.3, Theorem 1.4 and also Theorem 1.6. We emphasize that in our results from Sects 2, 3, 4, 5 and 6, X is a fixed smooth vector field in a domain \({\Omega }\) and satisfies the Hörmander condition; and that the Hamiltonian H(x, p) always enjoys (H0)-(H3). In all results from Sects. 5 and 6, we further assume \(\lambda _H=0\).
In Sect. refsps1, we state several facts about the analysis and geometry in Carnot-Carathéodory spaces employed in the proof.
In Sect. 3, we prove (i)\(\Leftrightarrow \)(ii)\(\Leftrightarrow \)(iii) in Theorem 1.3. Since (i)\(\Rightarrow \)(ii) follows from the definition and that (ii)\(\Rightarrow \)(iii) is obvious, it suffices to prove (iii)\(\Rightarrow \)(i). To this end, we borrow some ideas from [22, 36, 44, 45], which were designed for nonlocal Dirichlet energy forms originally.
The key is that, by employing assumptions (H0), (H1) and Mazur’s theorem, we are able to prove that if \(v_j\in \dot{W}^{1,\infty }_X(\Omega )\) with \(\Vert H(\cdot , Xv_j)\Vert _{L^\infty (\Omega )}\le \lambda \) and \(v_j\rightarrow v\) in \(C^0(\Omega )\) as \(j\rightarrow \infty \), then \(v \in \dot{W}^{1,\infty }_X(\Omega )\) with \(\Vert H(\cdot , Xv)\Vert _{L^\infty (\Omega )}\le \lambda \); see Lemma 3.1 for details. Thanks to this, choosing a suitable sequence of approximation functions via the definition of \(d_\lambda \), we then show that
See Lemma 3.3. Given any u satisfying (iii), we construct approximation functions \(u_j\) from \(d_\lambda \) and use Lemma 3.4 to show \(H(x,Du_j)\le \lambda \). That is (i) holds.
In Sect. 4, we prove (iii) in Theorem 1.3\(\Leftrightarrow \) (iv) in Theorem 1.4. Since (iii)\(\Rightarrow \)(iv) is obvious, it suffices to show (iv)\(\Rightarrow \)(iii).
This follows from the pseudo-length property of pseudo metric \(d_\lambda \) in Proposition 4.1. To get such a length property we find some special functions which fulfill the assumption Theorem 1.3(iii), and hence we show that the pseudo metric \(d_\lambda \) has a pseudo-length property.
In Sect. 5, we introduce McShane extensions and minimizers, and then gather several properties of them and pseudo-distance in Lemma 5.2 to Lemma 5.9, which are required in Sect. 6. These properties also have their own interests.
Given any domain \(U\Subset {\Omega }\), via the intrinsic distance \(d_\lambda ^U\) induced from U, we introduce McShane extensions \(\mathcal {S}_{g;V}^\pm \) of any \(g\in {\mathrm {\,Lip}}_{d_{CC}}(\partial U)\) in U. There are several reasons to use \(d^U_\lambda \) other than \(d_\lambda \), for example, \(d^U_\lambda \) has the pseudo-length property in U but the restriction of \(d_\lambda \) may not have; moreover, Theorem 1.3, and Theorem 1.4 holds if \((\Omega ,d_\lambda )\) therein is replaced by \( (U,d^U_\lambda )\), but not necessarily hold if \((\Omega ,d_\lambda )\) therein is replaced by \( (U,d _\lambda )\). However, the use of \(d_\lambda ^U\) causes several difficulties. For example, \(d_\lambda ^U\) may be infinity when extended to \(\overline{U}\). This makes it quite implicit to see the continuity of McShane extensions around \(\partial U\) from the definition. In Lemma 5.6, we get such continuity by analyzing the behaviour of \(d_\lambda ^U\) near \(\partial U\). Moreover, as required in Sect. 6, we have to study the relations between \(d_\lambda ^U\) and \(d^V_\lambda \) for subdomains V of U in Lemma 5.3 and Lemma 5.4.
In Sect. 6, we prove Theorem 1.6 in a constructive way by using above Rademacher type property and Perron’s approach, where we borrow some ideas from [9, 12, 14].
The proof consists of crucial Lemma 6.2, Lemma 6.4 and Proposition 6.5. Lemma 6.2 says that McShane extensions \(\mathcal {S}_{g;U}^\pm \) in U of function g in \(\partial U\) are local super/sub minimizers in U. Since \(\mathcal {S}_{g;U}^\pm \) are the maximum/minimum minimizers, the proof of Lemma 6.2 is reduced to showing that for any subdomain \(V \subset U\), the McShane extensions \({\mathcal S}^\pm _{h^\pm , V}\) in V with boundary \(h^\pm =\mathcal {S}_{g;U}^\pm |_{\partial V}\) satisfy
see Lemma 5.8 and Lemma 5.9 and the proof of Lemma 6.2. However, since Lemma 5.6 only gives
we must improve \(d_\lambda ^V(x,y)\) here to the smaller one \(d_\lambda ^U(x,y)\). To this end, we show \(d_\lambda ^V= d_\lambda ^U\) locally in Lemma 5.3, and also use the pseudo-length property of \( d_\lambda ^U\) heavily. Lemma 6.4 says that a function which is both of local superminimizers and subminimizer must be an absolute minizimzer. To get the required local minimizing property, we use McShane extensions to construct approximation functions and also need the fact \(d_\lambda ^V= d_\lambda ^{V\setminus \{x_i\}_{1\le i\le m}}\) in \(\overline{V}\times \overline{V}\) as in Lemma 5.4. Proposition 6.5 says that the supremum of local subminimizers are absolute minimizers. Due to Lemma 6.4, it suffices to prove the local super/sub minimizing property of such a supremum. We do prove this via using Lemma 6.2 and Lemma 5.9 repeatedly and a contradiction argument.
In Sect. 7, we aim at explaining some obstacles in using previous approach to establish the Rademacher type theorem and the existence of absolute minimizers. Indeed, in the literature to study Hamiltonians with better regularity or homogeneity, for instance, [14, 26], another intrinsic distance \(\bar{d}_\lambda \) is more common used in those proof which is hard to fit our setting.
In Appendix, we revisit the Rademacher’s theorem in the Euclidean space to show that Theorem 1.3 and Corollary 1.5 are indeed an extension of the Rademacher’s theorem.
2 Preliminaries
In this section, we introduce the background and some known results related to Carnot-Carathéodory spaces employed in the proof.
Let \(X:=\{X_1, ... ,X_m\}\) for some \(m\le n\) be a family of smooth vector fields in \({\Omega }\) which satisfies the Hörmander condition, that is, there is a step \(k \ge 1\) such that, at each point, \(\{X_i\}^{m}_{i=1}\) and all their commutators up to at most order k generate the whole \({{{\mathbb R}}^n}\). Then for each \(i=1,\cdots ,m\), \(X_i\) can be written as
with \(b_{il} \in C^\infty (\Omega )\) for all \(i=1,\cdots ,m\) and \(l=1,\cdots ,n\).
Define the Carnot-Carathéodory distance corresponding to X by
Here and below, we write \({\gamma }\in \mathcal {ACH}(0,1;x,y;\Omega )\) if \({\gamma }:[0,1]\rightarrow \Omega \) is absolutely continuous, \({\gamma }(0)=x,{\gamma }(1)=y\), and there exists measurable functions \(c_{i}:[0,1] \rightarrow {\mathbb R}\) with \(1 \le i \le m\) such that \(\dot{\gamma }(t) = \sum _{i=1}^m c_i(t)X_i({\gamma }(t))\) whenever \(\dot{\gamma }(t)\) exists. The length of \({\gamma }\) is
In the Euclidean case, we have the following remark.
Remark 2.1
In Euclidean case, that is, \(X=\{\frac{\partial }{\partial {x_i}}\}_{1\le i\le n}\), one has \(d_{CC}\) coincides with the intrinsic distance \(d_E^\Omega \) as given in (A.2). In particular, \(d_{CC}(x,y)=d^\Omega _E(x,y)\) for all \( x,y\in \Omega \) with \(|x-y|<{\mathrm {\,dist\,}}(x,\partial \Omega )\). When \(\Omega \) is convex, one further have and \(d_{CC}(x,y)=|x-y|\) for all \(x,y\in \Omega \); however, when \(\Omega \) is not convex, this is not necessarily true. See Lemma A.4 in Appendix for more details.
Since X is a Hörmander vector field in \(\Omega \), for any compact set \(K \subset \Omega \), there exists a constant \(C(K) \ge 1\) such that
see for example [41] and [27, Chapter 11]. This shows that the topology induces by \((\Omega ,d_{CC})\) is exactly the Euclidean topology.
Given a function \(u \in L^1_{{\mathrm {\,loc\,}}}(\Omega )\), its distributional derivative along \(X_i\) is defined by the identity
where \(X_i^*= -\sum _{l=1}^n \frac{\partial }{\partial x_l}(b_{il} \cdot )\) denotes the formal adjoint of \(X_i\). Write \(X^*= (X_1^*, \cdots ,X_m^*)\). We call \(Xu:=(X_1u, \cdots ,X_m u)\) the horizontal distributional derivative for \(u \in L^1_{{\mathrm {\,loc\,}}}(\Omega )\) and the norm |Xu| is defined by
For \(1\le p\le \infty \), denote by \(\dot{W}^{1,p}_X(\Omega )\) the p-th integrable horizontal Sobolev space, that is, the collection of all functions \(u\in L^1_{\mathrm {\,loc\,}}(\Omega )\) with its distributional derivative \(Xu\in L^p(\Omega )\). Equip \(\dot{W}^{1,p}_X(\Omega )\) with the semi-norm \(\Vert u\Vert _{\dot{W}^{1,p}_X(\Omega )}=\Vert |Xu|\Vert _{L^p(\Omega )}\).
The following was proved in [23, Lemma 3.5 (II)].
Lemma 2.2
If \( u \in \dot{W}^{1,p}_X(U)\) with \(1\le p<\infty \) and \( U\Subset \Omega \), then \(u^+=\max \{u,0\} \in \dot{W}^{1,p}_X(U)\) with \(Xu^+=(Xu)\chi _{\{x\in U|u>0\}}\) almost everywhere.
We recall the following imbedding of horizontal Sobolev spaces from [24, Theorem 1.4]. For any set \(U\subset \Omega \), the Lipschitz class \({\mathrm {\,Lip}}_{d_{CC}}(U)\) is the collection of all functions \(u:U\rightarrow {\mathbb R}\) with its seminorm
Theorem 2.3
For any subdomain \(U \Subset \Omega \), if \(u \in \dot{W}^{1,\infty }_X(U)\), then there is a continuous function \(\widetilde{u} \in {\mathrm {\,Lip}}_{d_{CC}}(U)\) with \(\widetilde{u}=u\) almost everywhere and
Remark 2.4
For any \(u \in \dot{W}^{1,\infty }_X(U)\), we call above \(\widetilde{u}\) given in Theorem 2.3 as the continuous representative of u. Up to considering \(\widetilde{u} \), in this paper we always assume that u itself is continuous.
We have the following dual formula of \({d_{CC}}\).
Lemma 2.5
For any \(x,y\in \Omega \), we have
To prove this we need the following bound for the norm of horizontal derivative of smooth approximation of functions in \(\dot{W}^{1,\infty }_{X}(\Omega )\), see for example [27, Proposition 11.10]. Denote by \(\{\eta _\epsilon \}_{\epsilon \in (0,1)}\) the standard smooth mollifier, that is, \(\eta _\epsilon (x) = \epsilon ^{-n} \eta (\frac{x}{\epsilon }) \quad \forall x\in {{{\mathbb R}}^n}\), where \(\eta \in C^\infty ({{{\mathbb R}}^n})\) is supported in unit ball of \({{{\mathbb R}}^n}\) (with Euclidean distance), \(\eta \ge 0\) and \(\int _{{{{\mathbb R}}^n}} \eta \,dx=1\).
Proposition 2.6
Given any compact set \(K\subset \Omega \), there is \(\epsilon _K\in (0,1)\) such that for any \(\epsilon <\epsilon _K\) and \(u \in \dot{W}^{1,\infty }_{X}(\Omega )\) one has
where \(A_\epsilon (u)\ge 0\) and \(\lim _{\epsilon \rightarrow 0}A_\epsilon (u) \rightarrow 0\) in K.
Proof of Lemma 2.5
Recall that it was shown by [32, Proposition 3.1] that
It then suffices to show that for any \(u\in \dot{W}^{1,\infty }_X(\Omega ) \) with \( \Vert |Xu|\Vert _{L^\infty (\Omega )}\le 1\), we have
Note that u is assumed to be continuous as in Remark 2.4.
To this end, given any \(x,y\in \Omega \), for any \( \epsilon >0\) there exists a curve \( {\gamma }_\epsilon \subset \mathcal {ACH}(0,1;x,y;\Omega )\) such that \( \ell _{d_{CC}}({\gamma }_\epsilon )\le (1+\epsilon )d_{CC}(x,y). \) We can find a domain \(U\Subset \Omega \) such that \({\gamma }_\epsilon \subset U\). It is standard that \(u*\eta _t\rightarrow u\) uniformly in \(\overline{U}\) and hence
Next, by Proposition 2.6, for \(0<t<t_{\overline{U}}\) one has
and moreover, \(A_t u(z) \rightarrow 0\) uniformly in \(\overline{U} \) as \(t \rightarrow 0\). Obviously, we can find \(t_{\epsilon ,\overline{U}}<t_{\overline{U}}\) such that for any \(0<t<t_{\epsilon ,\overline{U}}\), we have \(A_t u(x)\le \epsilon \) and hence, by \(\Vert |Xu|\Vert _{L^\infty (\Omega )}\le 1\), \(|X(u*\eta _t)(z)| \le 1+\epsilon ,\) for all \(z\in \overline{U}\). Therefore
Sending \(t\rightarrow 0\) and \(\epsilon \rightarrow 0\), one concludes \( u(y)-u(x)\le d_{CC}(x,y) \) as desired.\(\square \)
As a consequence of Rademacher type theorem (that is, Theorem 1.2), we have the following, which is an analogue of Lemma 6. Denote by \({\mathrm {\,Lip}}^*_{d_{CC}}(\Omega )\) the collection of all functions u in \(\Omega \)
with
Lemma 2.7
We have \( \dot{W}^{1,\infty }_X(\Omega )={\mathrm {\,Lip}}_{d_{CC}}(\Omega )={\mathrm {\,Lip}}^*_{d_{CC}}(u,\Omega )\) with
Proof
First, we show \({\mathrm {\,Lip}}_{d_{CC}}(\Omega )= {\mathrm {\,Lip}}^*_{d_{CC}}(\Omega )\) and \({\mathrm {\,Lip}}_{d_{CC}}( u,\Omega )= {\mathrm {\,Lip}}^*_{d_{CC}}( u,\Omega )\). Notice that \({\mathrm {\,Lip}}_{d_{CC}}(u,\Omega )\subset {\mathrm {\,Lip}}^*_{d_{CC}}(u,\Omega )\) and \({\mathrm {\,Lip}}^*_{d_{CC}}(u,\Omega ) \le {\mathrm {\,Lip}}_{d_{CC}}(u,\Omega )\) are obvious. We prove
Let \(u \in {\mathrm {\,Lip}}^*_{d_{CC}}(u,\Omega )\). Given any \(x,y\in \Omega \), and \({\gamma }\in \mathcal {ACH}(0,1;x,y;\Omega )\), parameterise \({\gamma }\) such that \(|\dot{\gamma }(t)| = \ell _{{d_{CC}}}({\gamma })\) for almost every \(t \in [0,1]\). Since
for each \(t\in [0,1]\) we can find \(r_t>0\) such that
Since \([0,1]\subset \cup _{t\in [0,1]}(t-r_t,t+r_t)\), we can find an increasing sequence \(t_i\in [0,1]\) with \(t_0 = 0\) and \(t_N=1\) such that
Write \(x_i={\gamma }(t_i)\) for \(i =0 , \cdots , N.\) We have
Noticing that \(A_{x,y} \le {\mathrm {\,Lip}}^*(u,\Omega ) < \infty \) for all \(x,y \in \Omega \), we deduce that
For any \(\epsilon >0\), recalling the definition of \({d_{CC}}\) in (2.1), there exists \(\{{\gamma }_\epsilon \}_{\epsilon >0} \subset \mathcal {ACH}(0,1;x,y;\Omega )\) such that
Combining (2.7) and (2.8), we have
Taking supremum among all \(x,y \in \Omega \) in the above inequality, we deduce that \(u \in {\mathrm {\,Lip}}_{d_{CC}}(u,\Omega )\) and \({\mathrm {\,Lip}}_{d_{CC}}(u,\Omega ) \le {\mathrm {\,Lip}}_{d_{CC}}^*(u,\Omega )\). Hence the second equality in (2.6) holds.
Next, we show \(\dot{W}^{1,\infty }_X(\Omega )={\mathrm {\,Lip}}_{d_{CC}}(\Omega )\) and \({\mathrm {\,Lip}}_{d_{CC}}(u,\Omega ) = \Vert | Xu| \Vert _{L^\infty (\Omega )}\). By Theorem 1.2, we know \({\mathrm {\,Lip}}_{{d_{CC}}}(\Omega ) \subset \dot{W}^{1,\infty }_X (\Omega )\) and \(\Vert | Xu| \Vert _{L^\infty (\Omega )} \le {\mathrm {\,Lip}}_{d_{CC}}(u,\Omega )\).
To see \(\dot{W}^{1,\infty }_X (\Omega ) \subset {\mathrm {\,Lip}}_{{d_{CC}}}(\Omega )\) and \({\mathrm {\,Lip}}_{d_{CC}}(u,\Omega ) \le \Vert | Xu| \Vert _{L^\infty (\Omega )}\), let \(u \in \dot{W}^{1,\infty }_X (\Omega )\). Then \(\Vert |Xu|\Vert _{L^\infty (\Omega )} =:\lambda < \infty .\) If \( \lambda >0\), then \(\lambda ^{-1}u \in \dot{W}^{1,\infty }_X (\Omega )\) and \(\Vert |X (\lambda ^{-1} u )|\Vert _{L^\infty (\Omega )} =1\). Hence \(\lambda ^{-1}u\) could be the test function in (2.2), which implies
or equivalently,
Therefore, \(u \in {\mathrm {\,Lip}}_{{d_{CC}}}(\Omega )\) and \( {\mathrm {\,Lip}}_{{d_{CC}}} (u,\Omega ) \le \Vert |Xu|\Vert _{L^\infty (\Omega )}\). If \(\lambda =0\), then similar as the above discussion, we have for any \(\lambda '>0\)
Therefore, \(u \in {\mathrm {\,Lip}}_{{d_{CC}}}(\Omega )\) and \( {\mathrm {\,Lip}}_{{d_{CC}}} (u,\Omega ) \le \lambda '\) for any \(\lambda '>0\). Hence \( {\mathrm {\,Lip}}_{{d_{CC}}} (u,\Omega ) =0 = \Vert |Xu|\Vert _{L^\infty (\Omega )}\) and we complete the proof.\(\square \)
Next, we recall some concepts from metric geometry. First we recall the notion of pseudo-distance.
Definition 2.8
We say that \(\rho \) is a pseudo-distance in a set \(\Omega \subset {{{\mathbb R}}^n}\) if \(\rho \) is a function in \(\Omega \times \Omega \) such that
-
(i)
\(\rho (x,x)=0\) for all \(x \in \Omega \) and \(\rho (x,y) \ge 0\) for all \(x,y \in \Omega \);
-
(ii)
\(\rho (x,z) \le \rho (x,y) + \rho (y,z)\) for all \(x,y,z \in \Omega \).
We call \((\Omega ,\rho )\) as a pseudo-metric space. The topology induced by \(\{\rho (x, \cdot )\}_{x \in \Omega }\) (resp. \(\{\rho (\cdot , x)\}_{x \in \Omega }\)) is the weakest topology on \(\Omega \) such that \(\rho (x, \cdot )\) (resp. \(\rho (\cdot , x)\)) is continuous for all \(x \in \Omega \).
We remark that since the above pseudo-distance \(\rho \) may not have symmetry, the topology induced by \(\{\rho (x, \cdot )\}_{x \in \Omega }\) in \(\Omega \) may be different from that induced by \(\{\rho (\cdot , x)\}_{x \in \Omega }\).
Suppose that H(x, p) is an Hamiltonian in \(\Omega \) satisfying assumptions (H0)-(H3). Let \(\{d_\lambda \}_{\lambda \ge 0}\) be defined as in (1.4). Thanks to the convention in Remark 2.4, one has
The following properties holds for \(d_\lambda \).
Lemma 2.9
The following holds.
-
(i)
For any \(\lambda \ge 0\), \(d_\lambda \) is a pseudo distance on \(\Omega \).
-
(ii)
For any \(\lambda \ge 0\),
$$\begin{aligned} R'_\lambda {d_{CC}}(x,y) \le d_\lambda (x,y) \le R_\lambda {d_{CC}}(x,y)<\infty \quad \forall x,y\in \Omega . \end{aligned}$$(2.10) -
(iii)
If \(H(x,p)=H(x,-p)\) for all \(p \in {\mathbb R}^m\) and almost all \(x \in \Omega \), then \(d_\lambda (x,y)=d_\lambda (y,x)\) for all \(x, y\in \Omega \).
Proof
To see Lemma 2.9 (i), by choosing constant functions as test functions in (2.9), one has \(\rho (x,y)\ge 0\) for all \(x,y\in \Omega \). Obviously, one has \(d_\lambda (x,x)=0\) for all \(x\in \Omega \). Besides,
By Definition 2.8, \(d_\lambda \) is a pseudo distance.
To see Lemma 2.9 (ii), by (H3), we have
From this and the definitions of \({d_{CC}}\) in (2.2) and \(d_\lambda \) in (2.9), we deduce (2.10) as desired.
Finally we show Lemma 2.9 (iii), since \(H(x,p)=H(x,-p)\) for all \(p \in {\mathbb R}^m\) and almost all \(x \in \Omega \), then
Hence for any \(x, y\in \Omega \), u can be a test function for \(d_\lambda (x,y)\) in the right hand side of (2.9) if and only if \(-u\) can be a test function for \(d_\lambda (y,x)\) in the right hand side of (2.9). As a result, \(d_\lambda (x,y)=d_\lambda (y,x)\), which completes the proof. \(\square \)
As a consequence of Lemma 2.9, we obtain the following.
Corollary 2.10
For any \(\lambda > \lambda _H\), \(d_\lambda \) is comparable with \(d_{CC}\), that is ,
Consequently, the topology induced by \(\{d_\lambda (x ,\cdot )\}_{x \in \Omega }\) and \(\{d_\lambda (\cdot ,x)\}_{x \in \Omega }\) coincides with the one induced by \({d_{CC}}\) in \(\Omega \), and hence, is the Euclidean topology.
Remark 2.11
(i) We remark that in Lemma 2.9 (iii), without the assumption \(H(x,p)=H(x,-p)\) for all \(p \in {\mathbb R}^m\) and almost all \(x \in \Omega \), \(d_\lambda \) may not be symmetric, that is, \(d_\lambda (x,y)=d_\lambda (y,x)\) may not hold for all \(x, y\in \Omega \).
(ii) If \(R'_\lambda =0\) for some \(\lambda >0\), then the topology induced by \(\{d_\lambda (x,\cdot )\}_{x \in \Omega }\) may be different from the Euclidean topology. To wit this, we construct an Hamiltonian H(p) in Euclidean disk \(\Omega =\{x\in {\mathbb R}^2||x|<1\}\) with \(X=\{\frac{\partial }{\partial x_1}, \frac{\partial }{\partial x_2} \}\), which satisfies (H0)-(H3) with \(\lambda _H>0\). Define \(H: \Omega \times {\mathbb R}^2 \rightarrow [0,\infty )\) by
where \(\chi _E\) is the characteristic function of the set E. One can check that H(p) satisfies (H0)-(H3). We omit the details.
Now we show
and thus \(\lambda _H \ge 1 >0\). Indeed, for any \( p=(p_1,p_2)\) and \(p_1\ge 0\), one has \(H(p )=|p|\) and hence \(H(p)\ge 1\) implies \(|p|\ge 1\). On the other hand, for any \( p=(p_1,p_2)\) and \(p_1< 0\), we always have \(H(p)=\max \{|p|,2\}\ge 2\), and hence
Writing \(e_1=(1,0)\), we claim that
This claim implies that the topology induced by \(\{d_1(x,\cdot )\}_{x\in \Omega }\) is different with the Euclidean topology.
To see the claim (2.12), writing \(o=(0,0)\), we only need to show that
It then suffices to show that \( u(ae_1)- u(0) \le 0 \) for all \(u \in W^{1,\infty }(\Omega ) \text{ with } \Vert H(\nabla u)\Vert _{L^\infty (\Omega )} \le 1\). Given such a function u, observe that \( \Vert H(\nabla u)\Vert _{L^\infty (\Omega )} \le 1\) implies \(\frac{\partial u}{\partial x_1}(x) \ge 0 \) and \(|\nabla u(x)|\le 1\) for almost all \(x \in {\mathbb R}^2\). Let \(\{{\gamma }_\delta \}_{0\le \delta \ll 1}\) be the line segment joining \(\delta e_2\) and \(ae_1+\delta e_2\) with \(e_2=(0,1)\), that is,
Since \(u \in W^{1,\infty }(\Omega )\) implies that u is ACL (see [28, Section 6.1]), there exists \(\{\delta _n\}_{n \in {\mathbb N}}\) depending on u such that u is differentiable almost everywhere on \({\gamma }_{\delta _n}\). Noting that \( \dot{\gamma }_{\delta _n}(t) =-e_1 \) and by \(\frac{\partial u}{\partial x_1}(x) \ge 0 \), one has
and hence
Thus
as desired.
Finally, we introduce the pseudo-length property.
Definition 2.12
We say a pseudo-metric space \((\Omega ,\rho )\) is a pseudo-length space if for all \(x,y \in \Omega \),
where \( \mathcal C(a, b; x, y; \Omega )\) denotes the class of all continuous curves \({\gamma }:[a,b]\rightarrow {\Omega }\) with \({\gamma }(a)=x\) and \({\gamma }(b)=y\), and
3 Proof of Theorem 1.3
In this section, we always suppose that the Hamiltonian H(x, p) enjoys assumptions (H0)-(H3). To prove Theorem 1.3, we first need several auxiliary lemmas.
Lemma 3.1
Suppose that \( \{u_j\}_{j\in {\mathbb N}}\subset \dot{W}^{1,\infty }_X(\Omega )\), and there exists \(\lambda \ge 0\) such that
If \(u_j\rightarrow u\) in \(C^0(\Omega )\), then \(u\in \dot{W}^{1,\infty }_X(\Omega )\) and \(\Vert H(x,Xu)\Vert _{L^\infty (\Omega )}\le \lambda \). Here and in below, for any open set \(V \subset \Omega \), \(u_j\rightarrow u\) in \(C^0(V)\) refers to for any \(K \Subset V\), \(u_j\rightarrow u\) in \(C^0(\overline{K})\).
Proof
By Lemma 2.7, one has
By (H3) and \(\Vert H(\cdot ,Xu_j)\Vert _{L^\infty (\Omega )}\le \lambda \), we have \(\Vert |Xu_j|\Vert _{L^\infty (\Omega )}\le R_\lambda \) for all j, and hence
that is, \( u\in {\mathrm {\,Lip}}_{d_{CC}}(\Omega )\). By Lemma 2.7 again, we have \(u\in {\dot{W}^{1,\infty }_X(\Omega )}\).
Next we show that \(\Vert H(x,Xu)\Vert _{L^\infty (\Omega )}\le \lambda \). It suffices to show that \(\Vert H(x,X(u|_U))\Vert _{L^\infty (U )}\le \lambda \) for any \(U\Subset \Omega \). Given any \(U\Subset \Omega \), we claim that \(X(u_j|_{ U })\) converges to \(X(u|_{ U })\) weakly in \(L^2( U ,{\mathbb R}^m)\), that is, for all \(1\le i\le m\), one has
To see this claim, note that \(\Vert |Xu_j|\Vert _{L^\infty (\Omega )}\le R_\lambda \) for all \(j\in {\mathbb N}\), and hence \(\Vert |Xu_j|\Vert _{L^2( U )}\le R_\lambda {| U |^{1/2}}\) for all \(j\in {\mathbb N}\). In other words, for each \(k\in {\mathbb N}\), the set \(\{X(u_{j }|_{ U })\}_{j\in {\mathbb N}}\) is bounded in \(L^2( U ,{\mathbb R}^m)\). By the weak compactness of \(L^2( U ,{\mathbb R}^m)\), any subsequence of \(\{Xu_{j }\}_{j\in {\mathbb N}}\) admits a subsubsequence which converges weakly in \(L^2( U ,{\mathbb R}^m)\). Therefore, to get the above claim, by a contradiction argument we only need to show that for any subsequence \(\{Xu_{j_s}\}_{s\in {\mathbb N}}\) of \(\{Xu_{j }\}_{j\in {\mathbb N}}\), if \( Xu_{j_s} \rightharpoonup q_k\) weakly in \(L^2( U , {\mathbb R}^m )\) as \(s\rightarrow \infty \), then \(X(u|_{ U }) =q_k\). For such \(\{Xu_{j_s}\}_{s\in {\mathbb N}}\), recalling that \(u_j\rightarrow u\) in \(C^0(\Omega )\) as \(j\rightarrow \infty \), for all \(1\le i \le m\) one has
for any \(\phi \in C^\infty _c( U )\). This implies that \(Xu|_{ U } =q_k\) as desired.
By Mazur’s Theorem, for any \(l>0\), we can find a finite convex combination \(w_l\) of \(\{X(u_j|_{ U })\}_{j=1}^{\infty }\) so that \(\Vert w_l-X(u|_{U})\Vert _{L^2( U )}\rightarrow 0 \) as \(l\rightarrow \infty \). Here \(w_l\) is a finite convex combination of \(\{X(u_j|_{ U })\}_{j=1}^{\infty }\) if there exist \(\{\eta _{j}\}_{j=1}^{k_l}\) for some \(k_l\) such that
By the quasi-convexity of \(H(x,\cdot )\) as in (H1), we have
Up to considering subsequence we may assume that \(w_l\rightarrow X(u|_{ U })\) almost everywhere in U. By the lower-semicontinuity of \(H(x,\cdot )\) as in (H0), we conclude that
The proof is complete. \(\square \)
Lemma 3.2
If \(v \in \dot{W}^{1,\infty }_{X }(\Omega )\), then
Consequently, let \(\{v_i\}_{1\le i\le j} \subset \dot{W}^{1,\infty }_{X }(\Omega )\) for some \(j\in {\mathbb N}\). If \( u=\max _{1\le i \le j}\{v_i\}\) or \( u=\min _{1\le i \le j}\{v_i\}\), then
Proof
First we prove (3.1). Let \(v \in \dot{W}^{1,\infty }_{X }(\Omega )\). By Lemma 2.7, \(v \in {\mathrm {\,Lip}}_{{d_{CC}}}(\Omega )\). Observe that
that is, \(v^+\in {\mathrm {\,Lip}}_{{d_{CC}}}(\Omega )\). By Lemma 2.7 again, \(v^+ \in \dot{W}^{1,\infty }_{X }(\Omega )\). To get \(Xv^+=(Xv)\chi _{\{x\in \Omega , v>0\}}\) almost everywhere, it suffices to consider the restriction \(v|_U\) of v in any bounded domain \(U \Subset \Omega \), that is, to prove \(X(v|_U)^+=(Xv|_U)\chi _{\{x\in U, v>0\}}\) almost everywhere. But this always holds thanks to Lemma 2.2 and the fact \(v|_U \in \dot{W}^{1,p}_{X}(U)\) for any \(1 \le p < \infty \).
Next we prove (3.2). If \(u =\max \{v_1,v_2\}\), where \(v_i \in \dot{W}^{1,\infty }_{X }(\Omega )\) for \(i=1,2\), then \(u =v_2+(v_1-v_2)^+.\) By (3.1), \(u \in \dot{W}^{1,\infty }_{X }(\Omega )\) and
Thus
A similar argument holds for \(u=\min \{v_1,v_2\}\). This gives (3.2) when \(j=2\). By an induction argument, we get (3.2) for all \(j\ge 2\). \(\square \)
Lemma 3.3
For any \( \lambda \ge 0\) and \(x \in \Omega \), we have \( d_\lambda (x, \cdot ), d_\lambda (\cdot ,x)\in \dot{W}^{1,\infty }_{X }(\Omega )\) and
Proof
Given any \(x\in \Omega \) and \(\lambda \ge 0\), write \(v(z)=d_\lambda (x,z)\) for all \(z\in \Omega \). To see \(H(\cdot ,Xv)\le \lambda \) almost everywhere, by Lemma 3.1, it suffices to find a sequence of function \(u_j\in { \dot{W}_X^{1,\infty }(\Omega )}\) so that \(H(\cdot ,Xu_j)\le \lambda \) almost everywhere and \(u_j\rightarrow v\) in \(C^0(\Omega )\) as \(j\rightarrow \infty \).
To this end, let \(\{K_j\}_{j \in {\mathbb N}}\) be a sequence of compact subsets in \(\Omega \) with
For \(j \in {\mathbb N}\) and \(y \in K_j\), by definition of \(d_\lambda \) we can find a function \(v_{y,j} \in \dot{W}^{1,\infty }_X(\Omega )\) such that \(H(\cdot ,Xv_{y,j})\le \lambda \) almost everywhere,
Since Lemma 2.9 implies that \(d_\lambda (x, \cdot )\) is continuous, there exists an open neighbourhood \(N_{y,j}\) of y with
Thanks to the compactness of \(K_j\), there exist \(y_1, \cdots , y_l\in K_j\) such that \(K_j \subset \bigcup _{i=1}^{l} N_{y_i,j}\). Write
Then \(u_j(x)=0\), and
Moreover by Lemma 3.2 we have
Since
is clear, we conclude that \(u_j\rightarrow {v} \) in \(C^0(K_i)\) as \(j\rightarrow \infty \) for all i, and hence \(u_j\rightarrow {v} \) uniformly in any compact subset of \(\Omega \) as \(j\rightarrow \infty \).
Similarly, we can show \(\Vert H(x,-Xd_\lambda (\cdot , x ))\Vert _{L^\infty (\Omega )} \le \lambda \) which finishes the proof.\(\square \)
In general, for any \(E \subset \Omega \), we define
Lemma 3.4
For any set \(E\subset \Omega \), we have \(d_{\lambda ,E} \in \dot{W}^{1,\infty }_{X }(\Omega )\) and \(\Vert H(x,Xd_{\lambda ,E} )\Vert _{L^\infty (\Omega )}\le \lambda \).
Proof
Let \(\{K_j\}_{j \in {\mathbb N}}\) be a sequence of compact subsets in \(\Omega \) with \(\Omega = \bigcup _{j \in {\mathbb N}} K_j\) and \(K_j \subset K_{j+1}^{\circ }\). For each j and \(y\in K_j\), we can find \(z_{y,j}\in E\) such that
Thus there exists a neighborhood N(y) of y such that
So we can find \(\{y_1,\cdots , y_{l_j}\}\) such that \(K_j=\cup _{i=1}^{l_j}N(y_i) \) for all \(i=1,\cdots l_j\). Write
This means that \(u_j\rightarrow d_{\lambda ,E}\) in \(C^0(\Omega )\) as \(j\rightarrow \infty \). Note that
By Lemma 3.1 we have \(d_{\lambda ,E}\in \dot{W}^{1,\infty }_X(\Omega )\) and \(\Vert H(\cdot , Xd_{\lambda ,E})\Vert _{L^\infty (\Omega )}\le \lambda \) as desired. \(\square \)
We are able to prove (i)\(\Rightarrow \)(ii)\(\Rightarrow \)(iii)\(\Rightarrow \)(i) in Theorem 1.3 as below.
Proof
(Proofs of (i)\(\Rightarrow \)(ii)\(\Rightarrow \)(iii)\(\Rightarrow \)(i) in Theorem 1.3) The definition of \(d_\lambda \) directly gives (i)\(\Rightarrow \)(ii). Obviously, (ii)\(\Rightarrow \)(iii). Below we prove (iii)\(\Rightarrow \) (i). Recall that (iii) says that \(u(y)-u(z)\le d_\lambda (z,y)\) for all \(x \in \Omega \) and \(y,z \in N(x)\), where \(N(x)\subset \Omega \) is a neighbourhood of x. To get (i), since \(\Omega =\cup _{x\in \Omega }N(x)\), it suffices to show that for all \(x\in \Omega \), one has \(u\in W_{X}^{1,\infty }(N(x))\) and \(\Vert H(\cdot ,Xu)\Vert _{L^\infty (N(x))}\le \lambda \).
Fix any x, and write \( U=N(x)\). Without loss of generality we assume that U is bounded.
Notice that \(u\in L^\infty (U)\). Let \(M\in {\mathbb N}\) so that \(M\ge \sup _{U} |u|\). For each \(k \in {\mathbb N}\) and \(l \in \{ -Mk, \cdots , Mk\}\), set
where
By Lemma 3.4, one has
Set
To get \(u\in \dot{W}_{X}^{1,\infty }(U)\ \text {and} \ \Vert H(\cdot ,Xu)\Vert _{L^\infty (U)}\le \lambda \), thanks to Lemma 3.1 with \(\Omega =U\), we only need to show \(u_k \rightarrow u\) in \(C^0(U)\) as \(k\rightarrow \infty \).
To see \(u_k \rightarrow u\) in \(C^0(U)\) as \(k\rightarrow \infty \), note that, for any \(k \in {\mathbb N}\), \(-M\le u \le M\) in U implies \(-Mk\le ku \le Mk\) in U. Thus, at any \(z \in U\), we can find \(j\in {\mathbb N}\) with \(-k\le j\le k\), which depends on z, such that \(Mj\le ku(z)\le M(j+1)\). Letting \(l=Mj\), we have \( \frac{l}{k}\le u(z)\le \frac{l+M}{k}\). We claim that \(u_k(z) \in [u(z),\frac{l+M}{k}]\). Obviously, this claim gives
and hence, the desired convergence \(u_k \rightarrow u\) in \(C^0(U)\) as \(k\rightarrow \infty \).
Below we prove the above claim \(u_k(z) \in [u(z),\frac{l+M}{k}]\). Recall that \(u(z)\in [\frac{l}{k},\frac{l+M}{k}]\) for some \(l =Mj\) with \(-k\le j\le k\).
First, we prove \(u_k(z) \le \frac{l+M}{k}\). If \(l+M > Mk\), then \(M < (l+M)/k\). Since
we have \(d_{\lambda ,F_{k,Mk}} (z)=0\) and hence,
Therefore,
If \(l+M\le Mk\), then \(u(z)\in [\frac{l}{k},\frac{l+M}{k}]\) implies \(z \in F_{k,l+M}\) and hence \(d_\lambda (z,F_{k,l+M})=0\). Thus
Combining (3.4) and (3.3), we have \( u_k(z) \le \frac{l+M}{k} \) as desired.
Next, we prove \(u(z)\le u_k(z)\). For any \(-k\le j\le k\) with \(Mj\le l\), since \(u(z) \ge \frac{l}{k} \ge \frac{Mj}{k}\), we can find \(w \in \partial F_{k,Mj}\) such that \(d_{\lambda ,F_{k,Mj}}(z)=d_\lambda (w,z)\). Since \(w \in \partial F_{k,Mj}\), we deduce that \(u(w)=\frac{Mj}{k}\) and
Note that \(w \in \overline{U}\), hence there exists a sequence \(\{w_s\}_{s \in {\mathbb N}}\subset U\) such that \(w_s\rightarrow w\) as \(s\rightarrow \infty \).
Thus by the assumption (iii),
By the triangle inequality and \(d_\lambda (w_s,w)\le R_\lambda d_{CC}(w_s,w)\) given in Lemma 2.9, we then obtain
Combining (3.5) and (3.6), we have
On the other hand, for any \(-k\le j\le k\) with \(Mj>l\), we have
From this and (3.7), it follows that
as desired. \(\square \)
4 Proof of Theorem 1.4
In this section, we always suppose that the Hamiltonian H(x, p) enjoys assumptions (H0)-(H3).
To prove Theorem 1.4, we need to show that \((\Omega ,d_\lambda )\) is a pseudo-length space for all \(\lambda \ge \lambda _H\) in the sense of Definition 2.12. In other words, define
where we recall the pseudo-length \(\ell _{d_\lambda }({\gamma })\) induced by \(d_\lambda \) defined in Definition 2.12 and \(\lambda _H\) in (1.7). We have the following.
Proposition 4.1
For any \(\lambda \ge \lambda _H\), we have \(d _\lambda =\rho _\lambda \).
To prove Proposition 4.1, we need the following approximation midpoint property of \(d_\lambda \).
Proposition 4.2
For any \(\lambda \ge 0\), we have
Proof
We prove by contradiction. Suppose that (4.1) were not true. There exists \(x_0,y_0\in \Omega \) such that
for some \(\epsilon _0>0\).
Given any \(\delta \in (0,\epsilon _0)\), define \(f(z):=f_1(z)+f_2(z)\) with
We claim that f satisfies Theorem 1.3(iii), that is, for any \(z\in \Omega \), there is an open neighborhood N(z) such that
Assume the claim (4.3) holds for the moment. Since we have already shown the equivalence between (ii) and (iii) in Theorem 1.3, we know that f satisfies Theorem 1.3(ii), that is,
In particular,
On the other hand, we have \(f_1(x_0)=-(r_0-\delta )\) and \(f_2(y_0)=r_0-\delta \). Since (4.2) implies
and \(f_2(x_0)=0\) and \(f_1(y_0)=0\). Therefore,
By \(\delta <\epsilon _0\), one has
which contradicts to (4.4).
Finally we prove the above claim (4.3). Firstly, thanks to Lemma 3.2 and 3.3, \(H(x,Xf_1)\le \lambda \) and \(H(x,Xf_2)\le \lambda \) almost everywhere in \( \Omega \), and hence, by the definition of \(d_\lambda \),
Next, set
and
For any \(z \in \Lambda _1\), that is, \(d_\lambda (x_0,z)<r_0\), (4.2) implies \(d_\lambda (z,y_0) \ge r_0\). Consequently,
and hence, \(f(z)=f_1(z).\) Consequently,
Similarly, for any \(z \in \Lambda _2\), that is, \(d_\lambda (z,y_0)<r_0\), (4.2) implies \(d_\lambda (x_0,z) \ge r_0\). Consequently,
and hence, \( f(z)= f_2(z).\) Consequently,
For any \(z\in \Lambda _3\), that is, \(d_\lambda (x_0,z)> r_0-\delta \) and \( d_\lambda (z,y_0)> r_0-\delta \), we have \(f_1(z)=0=f_2(z)\) and hence \(f(z) =0.\) Consequently
Noticing that \(\{\Lambda _i\}_{i=1,2,3}\) forms an open cover of \(\Omega \), for any \(z \in \Omega \), we choose
From (4.5), (4.6) and (4.7), we obtain (4.3) as desired with the choice of N(z) in (4.8). The proof is complete. \(\square \)
Lemma 4.3
Given any \(x,y\in \Omega \), the map \(\lambda \in [\lambda _H ,\infty )\mapsto d_\lambda (x,y)\in [0,\infty )\) is nondecreasing and right continuous.
Proof
The fact that \(d_\lambda (x,y)\) is non-decreasing with respect to \(\lambda \) is obvious for any \(x,y \in \Omega \) from the definition of \(d_\lambda \). Given any \(x,y \in \Omega \), we show the right-continuity the map \(\lambda \in [\lambda _H ,\infty )\mapsto d_\lambda (x,y)\in [0,\infty )\). We argue by contradiction. Assume there exists \(\lambda _0 \ge \lambda _H\) and \(x,y \in \Omega \) such that
Let \(w_\lambda (\cdot ):= d_\lambda (x,\cdot )\). By Lemma 3.3, we know \(\Vert H(\cdot ,Xw_\lambda )\Vert _{L^\infty (\Omega )}\le \lambda \). Since \(\{w_\lambda \}_{\lambda >\lambda _0}\) is a non-decreasing sequence with respect to \(\lambda \), \(\{w_\lambda \}\) converges pointwise to a function w as \(\lambda \rightarrow \lambda _0+\) and for any set \(V \Subset \Omega \) and \(x,y \in \overline{V}\), we have \(w_\lambda \rightarrow w\) in \(C^0(\overline{V})\). Then applying Lemma 3.1, we have
for any \(\lambda >\lambda _0\), which implies
By the definition of w, we have
Combining (4.10) and (4.11) and applying Theorem 1.3, we have
which contradicts to (4.9). The proof is complete. \(\square \)
We are in the position to show
Proof of Proposition 4.1
We consider the cases \(\lambda >\lambda _H\) and \(\lambda =\lambda _H\) separately.
Case 1. \(\lambda >\lambda _H\). First, \(d_\lambda \le \rho _\lambda \) follows from the triangle inequality for \(d_\lambda \).
To see \(\rho _\lambda \le d_\lambda \), it suffices to prove that for any \(z\in \Omega \), the function \(\rho _\lambda (z,\cdot ):\Omega \rightarrow {\mathbb R}\) satisfies Theorem 1.3(iii), that is, for any \( x\in \Omega \) we can find a neighborhood N(x) of x such that
Indeed, since we have already shown the equivalence of (i) and (iii) in Theorem 1.3, (4.12) implies that \(\rho _\lambda (z,\cdot )\) satisfies Theorem 1.3(i), that is, \(\rho _\lambda (z,\cdot )\in \dot{W}^{1,\infty }_X(\Omega )\) and \(\Vert H(\cdot ,X\rho _\lambda (z,\cdot ))\Vert _{L^\infty (\Omega )}\le \lambda \). Taking \(\rho _\lambda (z,\cdot )\) as the test function in the definition of \(d_\lambda (z,x)\), one has
as desired.
To prove (4.12), let \(z\in \Omega \) be fixed. For any \(x\in \Omega \) and any \(t>0\), write
and
For any \(x\in \Omega \), letting \(r_x = \min \{\frac{R_\lambda '}{10} {d_{CC}}(x, \partial \Omega ),1\}\), by Corollary 2.10, we have
where \(R_\lambda '>0\) thanks to (H3). Write \(N(x)= B_{d_\lambda }^-(x, {r_x})\). Given any \(w, y \in N(x)\), it then suffices to prove \(\rho _\lambda (z,y) -\rho _\lambda (z,w)\le d_\lambda (w, y ) \). To this end, for any \(0< \epsilon < \frac{1}{2} d_\lambda (w, y)\), we will construct a curve
Assume the existence of \({\gamma }_\epsilon \) for the moment. By the triangle inequality for \(\rho _\lambda \), we have
By sending \(\epsilon \rightarrow 0\), this yields \(\rho _\lambda (z,y) -\rho _\lambda (z,w)\le d_\lambda (w, y ) \) as desired.
Construction of a curve \({\gamma }_\epsilon \) satisfying (4.14). For each \(t \in {\mathbb N}\), set
We will use induction and Proposition 4.2 to construct a set
with \(y_0=w\) and \(y_1=y\) so that \(Y_t \subset Y_{t+1}\) , and that
The construction of \(\{Y_t\}_{t\in {\mathbb N}}\) is postponed to the end of this proof. Assuming that \(\{Y_t\}_{t\in {\mathbb N}}\) are constructed, we are able to construct \({\gamma }_\epsilon \) as below.
Firstly, set \(D:=\cup _{t\in {\mathbb N}} D_t\) and \(Y:=\cup _{t\in {\mathbb N}} Y_t\). Given any \(s_1, s_2 \in D\) with \(s_1 <s_2\), there exists \(t \in {\mathbb N}\) such that \(s_1=l2^{-t} \in D_{t}\), \(s_2=k2^{-t} \in D_{t}\) for some \(l < k\) and hence \(y_{s_1},y_{s_2} \in Y_t\). Using (4.16) and the triangle inequality for \(d_\lambda \), we have
Next, define a map \({\gamma }_\epsilon ^0 : D \rightarrow Y\) by \({\gamma }_\epsilon ^0(s) = y_s\) for all \(s \in D\). The above inequality (4.17) implies that
Since D is dense in [0, 1] and \(\overline{B_{d_\lambda }^+(y_0, {5r_x})}\) is complete, it is standard to extend \({\gamma }_\epsilon ^0\) uniquely to a continuous map \({\gamma }_\epsilon : [0, 1] \rightarrow B_{d_\lambda }^+(y_0, {6r_x})\), that is, \({\gamma }_\epsilon (s)={\gamma }_\epsilon ^0(s)\) for any \(s\in D\) and
Recalling (4.17), one therefore has
which gives \(\ell _{d_\lambda } ({\gamma }_\epsilon )\le \delta +\epsilon \). Thus the curve \({\gamma }_\epsilon \) satisfies (4.14) as desired.
Construction of \(\{Y_t\}_{t\in {\mathbb N}}\) via induction and Proposition 4.2. Since \(y,w \in B_{d_\lambda }^-(x,r_x)\), we have \(d_\lambda (w,x)<r_x\) and \( d_\lambda (x,y) < r_x\), which implies
We construct \(Y_1=\{y_0,y_{1/2},y_1\}\) which satisfies (4.16) with \(t=1\) and
We set \(y_0=w\) and \(y_1=y\). Noting that Proposition 4.2 gives
we choose \(y_{1/2}\in \Omega \) so that
Obviously, (4.19) gives (4.16). To see (4.18), obviously,
Moreover, noting that \(0<\epsilon<\frac{1}{2}\delta <r_x\) implies
and that \( y\in B_{d_\lambda }^-(x,r_x)\) implies \(d_\lambda (y,x)\le r_x\), we have
which gives \(y_{1/2} \in B_{d_\lambda }^+(x,3r_x).\)
In general, by induction given any \(t\ge 2\), assume that \(Y_{t-1}=\{y_s\}_{s\in D_{t-1}}\) is constructed so that
and that
Here and in what follows, we make the convention that \(\sum _{l=1}^{t-2}2^{-l} =0\) if \(t=2\). Since
the inclusion (4.20) implies \(Y_{t-1}\subset B_{d_\lambda }^+(x,5r_x)\) and hence (4.15).
Below, we construct \(Y_t=\{y_s\}_{s\in D_t}\) satisfying (4.16) and
Note that (4.22) and (4.23) imply \(Y_{t}\subset B_{d_\lambda }^+(x,5r_x)\) and hence (4.15).
We define \(Y_t=\{y_s\}_{s\in D_t}\) first. Since \(D_{t-1}\subsetneqq D_{t}\), for \(s \in D_{t-1}\), \(y_s \in Y_{t-1}\) is defined. It is left to define \(y_s\) for \(s\in D_t \setminus D_{t-1}\). Given any \(s\in D_t\setminus D_{t-1}\), we know that \(s=j2^{-t}\in D_t \setminus D_{t-1}\) for some odd j with \(1\le j\le 2^t-1\). Write \(s'=(j-1)2^{-t} \) and \(s''=(j+1)2^{-t} \). Then \(s',s''\in D_{t-1}\) and hence \(y_{s'}\in Y_{t-1}\) and \(y_{s''}\in Y_{t-1}\) are defined. Since Proposition 4.2 gives
we choose \(y_s\in \Omega \) such that
Note that (4.25) and (4.21) gives (4.16) directly. Indeed, for any \(0 \le j \le 2^t-1\), if j is odd, applying (4.25) with \(s=j2^{-t}\), \( s'=(j-1)2^{-t}\) and \(s''=(j+1)2^{-t}\), we deduce that
Since \(s',s''\in D_{t-1}\) and \(s''=s'+2^{-(t-1)}\), applying (4.21) to \(y_{s'},y_{s''}\) we have
If j is even, then \(j\le 2^t-2\). Applying (4.25) with \(s=(j+1)2^{-t}\), \( s'=j2^{-t}\) and \(s''=(j+2)2^{-t}\), we deduce that
Similarly, we also have (4.26).
To see (4.23), since (4.20) gives
it suffices to check
For any odd number j with \(1 \le j \le 2^t-1\), since \(y_{(j-1)2^{-t}} \in Y_{t-1}\), combining (4.26) and (4.27) and noting \(\epsilon<\delta <2r_x\), we obtain
which implies (4.23). We finish the proof of Case 1.
Case 2. \(\lambda =\lambda _H\). Fix \(x,y \in \Omega \). For any \(\epsilon >0\) sufficiently small, by the right continuity of the map \(\lambda \mapsto d_\lambda (x,y)\) at \(\lambda =\lambda _H\) from Lemma 4.3, there exists \(\mu >\lambda _H\) such that
By Case (i), there exists \({\gamma }:[0,1] \rightarrow \Omega \) joining x and y such that
By the definition of the pseudo-length and recalling from Lemma 4.3 that the map \(\lambda \mapsto d_\lambda (z,w)\) is non-decreasing for all \(z,w \in \Omega \), we have
Combining (4.28), (4.29) and (4.30), we conclude
The proof is complete.\(\square \)
We are ready to prove Theorem 1.4.
Proof of Theorem 1.4
Obviously, (iii) in Theorem 1.3\(\Rightarrow \) (iv) in Theorem 1.4. To see the converse, let \(\lambda \ge 0\) be as in (iv). Given any x and \(y,z \in N(x)\), where N(x) is given in (iv) we need to show
By Proposition 4.1, we know \((\Omega ,d_\lambda )\) is a pseudo-length space. Hence for any \(\epsilon >0\), there exists a curve \({\gamma }_\epsilon : [0,1] \rightarrow \Omega \) joining z and y such that
Since \({\gamma }_\epsilon \subset \Omega \) is compact, we can find a finite set \(\{t_i\}_{i=0}^n \subset [0,1]\) satisfying
where \(N({\gamma }_\epsilon (t_i))\) is the neighbourhood of \({\gamma }_\epsilon (t_i)\) in (iv). Hence by (iv) we have
Summing the above inequalities from 0 to \(n-1\), we have
where in the last inequality we applied (4.31). Letting \(\epsilon \rightarrow 0\) in the above inequality, we obtain (iii) in Theorem 1.3.
Finally, (1.8) is a direct consequence of (iv)\(\Leftrightarrow \)(i) and thanks to Lemma 4.3, the minimum in (1.8) is achieved.\(\square \)
Remark 4.4
The assumption \(R_\lambda '>0\) is needed in the proof of Proposition 4.1. Indeed, recall the construction of \({\gamma }_\epsilon \) in the proof of Proposition 4.1 below (4.17). To guarantee \({\gamma }_\epsilon \) is a continuous map, especially \({\gamma }_\epsilon ([0,1])\) is compact under the topology induced by \({d_{CC}}\), we need \(\{d_\lambda (x,\cdot )\}_{x \in \Omega }\) induces the same topology as the one by \({d_{CC}}\). By Corollary 2.10, \(R_\lambda '>0\) can guarantee this.
Moreover, to show \({\gamma }_\epsilon \Subset \Omega \) in (4.13) in the proof of Proposition 4.1, for each \(x \in \Omega \), we need the existence of \(r_x>0\), such that
Again, by Corollary 2.10, \(R_\lambda '>0\) can guarantee this. If \(R_\lambda '>0\) does not hold for some \(\lambda >0\), in Remark 2.11 (ii), the example shows that (4.32) may fail for some \(x \in \Omega \).
5 McShane extensions and minimizers
In this section, we always suppose that the Hamiltonian H(x, p) enjoys assumptions (H0)-(H3) and further that \(\lambda _H=0\).
Let \(U\Subset \Omega \) be any domain. Note that the restriction of \(d_\lambda \) in U may not have the pseudo-length property in U, and moreover, Theorem 1.3 with \(\Omega \) replaced by U may not hold for the restriction of \(d_\lambda \) in U. Thus instead of \(d_\lambda \), below we use intrinsic pseudo metrics \(\{d^U_\lambda \}_{\lambda > 0}\) in U, which are defined via (1.4) with \( \Omega \) replaced by U, that is,
Obviously we have proved the following.
Corollary 5.1
Theorem 1.3, Theorem 1.4 and Proposition 4.1 hold with \(\Omega \) replaced by U and \(d_\lambda \) replaced by \(d_\lambda ^U\). In particular, \(d_\lambda ^U\) has the pseudo-length property in U for all \(\lambda \ge 0\).
Observe that, apriori, \(d^U_\lambda \) is only defined in U but not in \(\overline{U}\). Naturally, we extend \(d^U_\lambda :U\times U\rightarrow [0,\infty )\) as a function \(\widetilde{d}^U_\lambda : \overline{U}\times \overline{U}\rightarrow [0,\infty ]\) by
Obviously, \(\widetilde{d}^U_\lambda (x,y)=d^U_\lambda (x,y)\) for all \((x,y)\in U\times U\), and \(\widetilde{d}_\lambda ^U\) is lower semicontinuous in \(\overline{U}\times \overline{U}\), that is, for any \(a\in {\mathbb R}\), the set
is open in \(\overline{U}\). One may also note that it may happen that \(d^{U}_\lambda (x,y)=+\infty \) for some \((x,y)\notin U\times U\). Below, for the sake of simplicity, we write \(\widetilde{d}_\lambda ^U\) as \(d_\lambda ^U\). We define \(d_{CC}^U\) by letting \(H(x,p)=|p|\) in U and \(\lambda =1\) in (5.1).
The following property will be used later.
Lemma 5.2
Let \(U \Subset \Omega \) be a subdomain and \(\lambda \ge 0\).
-
(i)
For any \(x,y\in \overline{U}\), we have \(d^{U}_\lambda (x,y) \ge d_\lambda (x,y)\ge R'_\lambda d_{CC}(x,y)\).
-
(ii)
For any \(x\in U\) and \(y \in U\) with \({d_{CC}}(x,y) < {d_{CC}}(x,\partial {U})\), we have \( d^{U}_\lambda (x, y)\le R_\lambda d_{CC}(x,y). \)
-
(iii)
For any \(x\in U\), let \(x^*\in \partial U\) be the point such that \( d_{CC}(x,x^*)=d_{CC}(x,\partial U)\). Then
$$\begin{aligned} d_\lambda ^U (x,x^*)\le R_\lambda d_{CC}(x,x^*)<\infty . \end{aligned}$$ -
(iv)
For any \(x\in \overline{U}\) and \( y \in U\) we have
$$\begin{aligned} d^{U}_\lambda (x, z) \le d^{U}_\lambda (x,y) + d^{U}_\lambda (y,z) \text{ and } d^U_\lambda (z,x) \le d^U_\lambda (z, y)+ d^U_\lambda (y,x) \quad \forall z\in \partial U. \end{aligned}$$ -
(v)
Given any \(z\in \partial U\), if \(d^{U}_\lambda (x,z)= \infty \) for some \(x\in \overline{U}\), then \(d^{U}_\lambda (y,z)=+\infty \) for any \(y \in \overline{U}\).
-
(vi)
Given any \(x,y\in \overline{U}\times \overline{U}\) the map \(\lambda \in [0,\infty )\mapsto d_\lambda ^U(x,y)\in [0,\infty ]\) is nondecreasing and for \({ 0<\lambda<\mu <\infty }\),
$$\begin{aligned} d^U_\lambda (x,y)<\infty \text { if and only if } d^U_\mu (x,y)<\infty . \end{aligned}$$As a consequence,
$$\begin{aligned} \overline{U}^*:=\{y\in \overline{U} \ | \ d_{\lambda }^U(x,y)<\infty \quad \text{ for } \text{ some } x\in U \text{ and } \lambda >0 \} \end{aligned}$$is well-defined independent of the choice of \(\lambda >0\).
-
(vii)
Given any \(x,y\in \overline{U}^*\times \overline{U}^*\), the map \(\lambda \in { [0 ,\infty )}\mapsto d_\lambda ^U(x,y)\in [0,\infty ]\) is right continuous.
Proof
To see (i), for any \(x,y\in U\), since the restriction \(u|_U\) is a test function in the definition of \(d^{U}_\lambda (x,y)\) whenever u is a test function in the definition of \(d_\lambda (x,y)\), we have \(d^U_\lambda (x,y)\ge d_\lambda (x,y)\). In general, given any \((x,y)\in (\overline{U}\times \overline{U})\setminus (U\times U)\), for any \(r>0\) sufficiently small, we have \(d^U_\lambda (z,w)\ge d_\lambda (z,w)\) whenever \(z,w\in U\) and \(|(z,w)-(x,y)|\le r\). By the continuity of \(d_\lambda \) in \(\Omega \times \Omega \), we have
that is, \( d_\lambda ^U(x,y)\ge d_\lambda (x,y)\). Recall that \(d_\lambda (x,y)\ge R'd_{CC}(x,y)\) comes from Lemma 2.1.
To see (ii), given any \(y\in U\) with \({d_{CC}}(x,y)<{d_{CC}}(x,\partial {U})\), there is a geodesic \({\gamma }\) with respect to \({d_{CC}}\) connecting x and y so that \({\gamma }\subset B_{d_{CC}}(x,d_{CC}(x,\partial U))\). For any function \(u\in \dot{W}^{1,\infty }_{X }(U)\) with \(\Vert H(\cdot ,Xu)\Vert _{L^\infty (U)}\le \lambda \), we know that \(\Vert Xu\Vert _{L^\infty (U)}\le R_\lambda \). Let \(U' \Subset U\) and \(x,y \in U'\). Thanks to Proposition 2.6, we can find a sequence \(\{u_k\}_{k\in {\mathbb N}} \subset C^\infty (U')\) such that \(u_k\rightarrow u\) everywhere as \(k\rightarrow \infty \) and \(\Vert Xu_k\Vert _{L^\infty (U')}\le R_\lambda + A_k(u)\) with \(\lim _{k \rightarrow \infty } A_k(u) \rightarrow 0\). Since
we have
Taking supremum in the above inequality over all such u, we have \(d^U_\lambda (x,y)\le R_\lambda {d_{CC}}(x,y)\).
To see (iii), given any \(x\in U\), there exists \(x^*\in \partial U\) such that \( d_{CC}(x,x^*)=d_{CC}(x,\partial U)\). By (ii) and the definition of \(d^U_\lambda (x,x^*)\), we know that \(d_\lambda ^U (x,x^*)\le R_\lambda d_{CC}(x,x^*)<\infty \).
To get (iv), for any \( x\in \overline{U}\), \(y\in U\) and \(z\in \partial U\), choose \(x_k,z_k\in U\) such that \(d^{U}_\lambda (x_k,y)\rightarrow d^{U}_\lambda (x,y)\) and \(d^{U}_\lambda (y,z_k)\rightarrow d^{U}_\lambda (y,z)\) as \(k\rightarrow \infty \). Since
letting \(k\rightarrow \infty \) and by the lower-semicontinuous of \(d^U_\lambda \) we get
In a similar way, we also have
Note that (v) is a direct consequence of (iv).
We show (vi). The fact that \(d_\lambda ^U(x,y)\) is non-decreasing with respect to \(\lambda \) is obvious for any \(x,y \in \overline{U}\). Assume \(0< \lambda<\mu <\infty \). Then \(d^U_\mu (x,y)<+\infty \) implies \(d^U_\lambda (x,y)<+\infty \). Conversely, if \(d_{\mu }^U(x,y)=+\infty \), we show \(d_\lambda ^U(x,y)=+\infty \). This may happen if at least one of x and y lie in \(\partial {U}\). Then for any \(\{x_k\}_{k\in {\mathbb N}} \subset U\) and \(\{y_k\}_{k\in {\mathbb N}}\subset U\) converging to x and y, it holds
where we let \(x_k \equiv x\) (resp. \(y_k \equiv y\)) if \(x \in U\) (resp. \(y\in U\)). By (i) and (ii), we have for any \(\lambda \le \mu \)
where we recall that \(R'_\lambda >0\) for all \(\lambda >0\).
Finally, we show (vii). Since we only consider \(x,y \in \overline{U}^*\), by an approximation argument, it is enough to show the right-continuity for \(x,y \in U\). The proof is similar to the one of Lemma 4.3. We omit the details. The proof of Lemma 5.2 is complete.\(\square \)
Lemma 5.3
Suppose that \( U \Subset \Omega \) and that V is a subdomain of U. For any \(\lambda \ge 0\), one has
Conversely, given any \(\lambda >0\) and \(x \in V\), there exists a neighborhood \(N _\lambda (x) \Subset V\) of x such that
Proof
For any \(u \in \dot{W}^{1,\infty }_{X }(U)\) with \(\Vert H(\cdot , Xu)\Vert _{L^\infty (U)} \le \lambda \), we know that the restriction \(u|_{V} \in \dot{W}^{1,\infty }_{X }(V)\) with \(\Vert H(\cdot , Xu|_V)\Vert _{L^\infty (V)} \le \lambda \). Hence by the definition of \(d^U_\lambda \) and \(d^V_\lambda \), \( d^U_\lambda (x,y) \le d^V_\lambda (x,y) \) for all \( x,y \in V \) and then for all \( x,y \in \overline{V}.\)
Conversely, we just show \(d^U_\lambda (x,\cdot ) = d^V_\lambda (x,\cdot )\) in some neighborhood N(x). In a similar way, we can prove \(d^U_\lambda (\cdot , x) = d^V_\lambda (\cdot ,x)\) in some neighborhood N(x).
By Lemma 5.2 (i), one has \(d_\lambda ^V (x,y) \ge R'_\lambda d_{CC} (x,y) \) for any \(x,y\in V\). Thus for any \(r>0\), \(d^V_\lambda (x,y)\le r\) implies \( d_{CC} (x,y)\le r/R'_\lambda \). Given any \(x\in V\) and \(0<r<{d_{CC}}(x,\partial V)/R'_\lambda \), we therefore have
Define \(u_{x,r}: \Omega \rightarrow {\mathbb R}\) by
If \( r< {d_{CC}}(x,\partial V)R'_\lambda /4\), we claim that
Assume claim (5.2) holds for the moment. By (5.2), we are able to take \(u_{x,r}|_{U}\) as a test function in \(d^U_\lambda \) so that
On the other hand, for any \(y\in N_r(x)\), since \(d^V _\lambda (x,y)= u_{x,r}(y)-u_{x,r}(x)\), we get \(d^V _\lambda (x,y)\le d^U_\lambda (x,y)\) as desired.
Finally, we prove the claim (5.2).
Proof of the claim (5.2). First, by Lemma 3.3 and Lemma 3.4, the restriction \(u_{x,r}|_V\) of \(u_{x,r}\) in V belongs to \(\dot{W}^{1,\infty }_X(V)\) with \(H(z,X u_{x,r}|_V(z))\le \lambda \) almost everywhere in V, and hence
Next, we show \(u \in {\mathrm {\,Lip}}_{d_{CC}}(\Omega )\). Given any \(w,z\in \Omega \), we consider 3 cases separately.
Case 1. \(w \in \overline{B_{d_{CC}}(x,r/R'_\lambda )}\) and \(z\in \overline{ B_{d_{CC}}(x, r /R'_\lambda )}\). We have \(z\in \overline{B_{d_{CC}}(w,2 r/R'_\lambda )}\). Since
by Lemma 5.2 (ii), \(d^V_\lambda (w,z)\le R_\lambda d_{CC}(w,z)\), which combined with (5.3), gives
Case 2. \(w,z\notin B_{d_{CC}}(x,r / R'_\lambda )\). Then \(w,z \in \Omega \setminus B_{d_{CC}}(x,r / R'_\lambda )\), since \(u_{x,r} \) is constant r in \(\Omega \setminus B_{d_{CC}}(x,r /R'_\lambda )\), we know that
Case 3. \(w \in \overline{B_{d_{CC}}(x,r/R'_\lambda )}\) and \(z\notin \overline{B_{d_{CC}}(x,r/R'_\lambda )}\). Then for any \(\epsilon >0\), there exists a curve \({\gamma }_\epsilon :[0,1] \rightarrow \Omega \) joining z and w such that
and there exists \(t \in [0,1]\) such that \({\gamma }_\epsilon (t) \in \partial B_{d_{CC}}(x,r/R'_\lambda )\). Thus using Case 1 and Case 2, we deduce
Letting \(\epsilon \rightarrow 0\) in the above inequality, we conclude
Finally, by Lemma 2.7, \(u_{x,r}\in \dot{W}^{1,\infty }_X(\Omega )\). Note that \(X u_{x,r}=0\) in \(\Omega \setminus \overline{B_{d_{CC}}(x,r/R'_\lambda )}\) implies \(H(z,X u_{x,r}(z) )=0\) almost everywhere. Therefore recalling \(H(z,X u_{x,r}(z) )\le \lambda \) almost everywhere in V and \(\Omega = V \cup (\Omega \setminus \overline{B_{d_{CC}}(x,r/R'_\lambda )})\), we conclude \(H(z,X u_{x,r}(z) )\le \lambda \) almost everywhere in \(\Omega \).\(\square \)
Lemma 5.4
Suppose that \( U \Subset \Omega \) and that \(V=U\setminus \{x_i\}_{1\le i\le m}\) for some \(m\in {\mathbb N}\) and \(\{x_i\}_{1\le i\le m}\subset U\). Then for any \(\lambda \ge 0\), one has
Proof
Obviously \(d_\lambda ^U\le d^V_\lambda \) in \(\overline{U}\). Conversely, we show \(d_\lambda ^V\le d^U_\lambda \) in \(\overline{U}\). First, by an approximation argument, it suffices to consider \(x,y \in U\). By the right continuity of \(\lambda \in [0,\infty )\rightarrow d_\lambda ^U(x,y)\) for any \(x,y\in \overline{U}^*\), up to considering \( d_{\mu +\epsilon }^U\) for sufficiently small \(\epsilon >0\), we may assume that \(\mu >0\). For any \(\lambda >0\), by the pseudo-length property of \(d^U_\lambda \) as in Proposition 4.1, it suffices to prove that for any curve \({\gamma }:[a,b] \rightarrow U\), one has
We consider the following 3 cases.
Case 1. \({\gamma }((a,b))\subset V\) and \({\gamma }(\{a,b\}) \subset V\), that is, \({\gamma }( [a,b])\subset V\). Recall from Lemma 5.3 that, for each \(x \in V\), there exists a neighborhood N(x) such that
Since \({\gamma }\subset \cup _{t \in [a,b]}N({\gamma }(t))\), we can find \(a=t_0=0<t_1<\cdots < t_m=b\) such that \({\gamma }\subset \cup _{i=0}^m N({\gamma }(t_i))\) and \({\gamma }([t_i,t_{i+1}])\subset N({\gamma }(t_i))\). By the triangle inequality, one has
Case 2. \({\gamma }((a,b))\subset V\) and \({\gamma }(\{a,b\})\not \subset V\). Applying Case 1 to \({\gamma }|_{[a+\epsilon ,b-\epsilon ]}\) for sufficiently small \(\epsilon >0\), we get
Case 3. \({\gamma }((a,b))\not \subset V\). Without loss of generality, for each \(x_i\), there is at most one \(t \in (a,b)\) such that \({\gamma }(t )=x_i\). Indeed, let \(t ^\pm \) as the maximum/mimimum \(s\in (a,b)\) such that \({\gamma }(s)=x_1\). Then \( a\le t ^-\le t^+ \le b\). If \(t ^-<t ^+ \), we consider \( {\gamma }_1:[a,b-(t^+ -t^- )]\rightarrow U\) with \({\gamma }_1(t)={\gamma }(t)\) for \(t\in [a,t ^-]\), and \({\gamma }_1(t)={\gamma }(t-(t^+ -t^- ))\) for \(t\in [t^+ ,b]\). Then \(\ell _{d^U_\lambda }({\gamma }_1) \le \ell _{d^U_\lambda }({\gamma }) \). Repeating this procedure for \(x_2,\cdots ,x_m\) in order, we may get a new curve \(\eta \) such that for each \(x_i\), there is at most one \(t\in (a,b)\) such that \({\gamma }(t )=x_i\) and \(\ell _{d^U_\lambda }(\eta )\le \ell _{d^U_\lambda }({\gamma }).\)
Denote by \(\{a_j\}_{j=0}^s\) with \(a=a_0<a_1<\cdots <a_s=b\) such that \({\gamma }( \{a_1,\cdots ,a_{s-1}\}) \subset U\setminus V=\{x_i\}_{1\le i\le m}\) and \({\gamma }([a,b]\setminus \{a_1,\cdots ,a_{s-1}\})\subset V\). Applying Case 2 to \({\gamma }|_{[a_j,a_{j+1}]}\) for all \(0\le j\le s-1\), we obtain
as desired. The proof is complete.\(\square \)
For any \(g\in C^0(\partial U)\), write
The following lemma says the infimum can be reached.
Lemma 5.5
We have
Proof
First, if \(x \in \partial {U}\setminus \overline{U}^*\) or \(y \in \partial {U}\setminus \overline{U}^*\), we have \(d^{U}_\lambda (x, y) = \infty \). Hence \(g(y) - g(x) \le d^{U}_\lambda (x, y)\) holds trivially, which implies that
Thanks to Lemma 5.2(vii), we finish the proof.\(\square \)
If \(\mu =\mu (g,\partial U)<\infty \), we define
Note that \({\mathcal S}^\pm _{g; U}\) serve as “McShane" extensions of g in U.
Lemma 5.6
If \(\mu =\mu (g,\partial U)<\infty \), then we have
-
(i)
\(\mathcal {S}_{g;U}^\pm \in \dot{W}^{1,\infty }_X(U)\cap C^0(\overline{U})\) with \(\mathcal {S}_{g;U}^\pm =g\) on \(\partial U\);
-
(ii)
for any \( x,y\in \overline{U}\),
$$\begin{aligned} \mathcal {S}_{g;U}^\pm (y)-\mathcal {S}_{g;U}^\pm (x)\le d^U_\mu (x,y) ;\end{aligned}$$(5.5) -
(iii)
\( \Vert H(\cdot ,{X\mathcal {S}_{g;U}^\pm } )\Vert _{L^\infty (U)}\le \mu .\)
Proof
By Corollary 5.1, (ii) implies (iii) and \( \mathcal {S}_{g;U}^\pm \in \dot{W}^{1,\infty }_X(U)\). Below we show \(\mathcal {S}_{g;U}^\pm =g\) on \(\partial U\), (ii) and \(\mathcal {S}_{g;U}^\pm \in C^0(\overline{U})\) in order.
Proof for \(\mathcal {S}_{g;U}^\pm =g\) on \(\partial U\). For any \(x\in \partial U\), by definition we have
Conversely, for \(y \in \partial {U}\), one has
and hence \(\mathcal {S}_{g;U}^-(x) \le g(x)\le \mathcal {S}_{g;U}^+(x) \) as desired.
Proof of (ii). We only prove (ii) for \(\mathcal {S}_{g;U}^-\); the proof for \(\mathcal {S}_{g;U}^+\) is similar. For \(x,y\in \partial U\), by the definition of \(\mu \) one has
For \(x\in U\) and \(y\in \partial U\), by definition
and hence
For \(x\in \overline{U}\) and \( y \in U\), by Lemma 5.2(iv), we have
One then has
and hence
Proof of \(\mathcal {S}_{g;U}^\pm \in C^0(\overline{U})\). We only consider \(\mathcal {S}_{g;U}^-\in C^0(\overline{U})\); the proof for \(\mathcal {S}_{g;U}^+\in C^0(\overline{U})\) is similar. It suffices to show that for any \(x \in \partial {U}\) and a sequence \(\{x_j\} \subset U\) converging to x,
Choosing \(x^*_j\in \partial U\) such that \(d_{CC}(x_j,x^*_j)={\mathrm {\,dist\,}}_{d_{CC}}(x_j,\partial U)\), one has
Thanks to Lemma 5.2(iii) with \(x=x_j\) and \(x^*=x^*_j\) therein, we deduce
Since
by the continuity of g, we have
Assume that
By the definition we can find \(z_j\in \partial U\) such that
Thus for \(j \in {\mathbb N}\) sufficiently large, we have
Up to some subsequence, we assume that \(z_j\rightarrow z \in \partial U\). Note that
By the continuity of g, we conclude
which is a contradiction with the definition of \(\mu \) and \(\mu <\infty \).\(\square \)
Write
A function \( u:\overline{U}\rightarrow {\mathbb R}\) is called as a minimizer for \(\textbf{I}(g,U)\) if
We have the following existence and properties for minimizers.
Lemma 5.7
For any \(g\in C(\partial U)\) with \(\mu (g,\partial U)<\infty \), we have the following:
-
(i)
We have \( \mu (g,\partial U)=\textbf{I}(g,U). \) Both of \(\mathcal {S}_{g;U}^\pm \) are minimizers for \(\textbf{I}(g,U)\).
-
(ii)
If u is a minimizer for \(\textbf{I}(g,U)\) then
$$\begin{aligned} \mathcal {S}_{g;U}^- \le u \le \mathcal {S}_{g;U}^+ \quad \text{ in }\quad \overline{U}\quad \hbox {and } \Vert H(x,Xu) \Vert _{L^\infty (U)} = \textbf{I}(g,U)=\mu (g,\partial U) . \end{aligned}$$ -
(iii)
If u, v are minimizer for \(\textbf{I}(g,U)\), then \(tu+(1-t)v\) with \(t\in (0,1)\), \(\max \{u, v\}\) and \(\min \{u, v\}\) are minimizers for \(\textbf{I}(g,U)\).
Proof
(i) Since \(\mu (g,\partial U) <\infty \), then by Lemma 5.6, we know that \(\mathcal {S}_{g;U}^\pm \) satisfies the condition required in (5.6) and hence
Below we show that \(\mu (g,\partial U)\le \textbf{I}(g,U) .\) Note that combining this and (5.7) we know that
and moreover, \(\mathcal {S}_{g;U}^\pm \) are minimizers for \({ \textbf{I}}(g;U)\).
For any \(\lambda >\textbf{I}(g,U)\), there is a function \(u\in \dot{W}^{1,\infty }_X(U)\cap C^0(\overline{U})\) with \(u=g\) on \(\partial U\) such that \( \Vert H(x,Xu)\Vert _{L^\infty (U)}\le \lambda \). By Corollary 5.1,
By the continuity of u in \(\overline{U}\) and the definition of \(d^U_\lambda \) in \(\overline{U}\times \overline{U}\) we have
Thus \(\mu (g,\partial U)\le \textbf{I}(g,U) .\)
(ii) If u is a minimizer for \({ \textbf{I}}(g,U)\), one has
By Corollary 5.1, \(u(y) - u(x) \le d_\mu ^U(x, y)\) for any \(x, y \in U\) and hence, by the continuity of u and the definition of \(d_\mu ^U\), for all \(x, y \in \overline{U}\). Since \(u= g\) on \(\partial U\), for any \(x \in U\), one has \(g(y) - d_\mu ^U(x, y) \le u(x)\) for any \(y \in \partial U\), which yields \(\mathcal {S}_{g;U}^-(x) \le u(x)\). By a similar argument, one also has \(u \le \mathcal {S}_{g;U}^+\) in U as desired.
(iii) Suppose that \(u_1,u_2\) are minimizers for \(\textbf{I}(g,U)\). Set
Then \(u_\eta \in \dot{W}^{1,\infty }_X(U)\cap C^0(\overline{U})\), and \(u_\eta = g\) on \(\partial U\), and by (H1),
This, combined with the definition of \(\textbf{I}(g,U)\), implies that \(u_\eta \) is also a minimizer for \({ \textbf{I}}(g,U)\).
Finally, note that
By Lemma 3.2, one has
We know that
that is, \(\max \{u_1, u_2\}\) is a minimizer for \({ \textbf{I}}(g,U)\). Similarly, \(\min \{u_1, u_2\}\) is a minimizer for \({ \textbf{I}}(g,U)\).\(\square \)
We have the following improved regularity for Mschane extension via \(d_\lambda \).
Lemma 5.8
Suppose that \(U\Subset \Omega \) and that \(V \subset U\) is a subdomain. If \(g:\partial V\rightarrow {\mathbb R}\) satisfies
for some \(\lambda \ge 0\), then \( \mu (g,\partial U)\le \lambda \) and
Proof
Since \(d_\lambda ^U\le d^V_\lambda \) in \(\overline{V}\times \overline{V}\), we know that
and hence
To prove (5.9), by the pseudo-length property of \(d_\lambda ^U\) as in Corollary 5.1, it suffices to prove that for any curve \({\gamma }:[a,b] \rightarrow \Omega \) with \({\gamma }(a),{\gamma }(b)\in \overline{V}\), one has
We consider the following 4 cases.
Case 1. \({\gamma }((a,b))\subset V\) and \({\gamma }(\{a,b\}) \subset V\), that is, \({\gamma }( [a,b])\subset V\). Noting \(\mu =\mu (g,\partial V)\le \lambda \), one then has \(d_\mu ^U \le d_\lambda ^V \) in \( V\times V\). From the definition of \(S^\pm _{g,V}\), it follows
Recall from Lemma 5.3 that, for each \(x \in V\), there exists a neighborhood \(N(x) \Subset V\) such that
We therefore have
Since \({\gamma }\subset \cup _{t \in [a,b]}N({\gamma }(t))\), we can find \(a=t_0=0<t_1<\cdots < t_m=b\) such that \({\gamma }\subset \cup _{i=0}^m N({\gamma }(t_i))\) and \({\gamma }([t_i,t_{i+1}])\subset N({\gamma }(t_i))\). Applying (5.11) to \({\gamma }(t_i)\) and \({\gamma }(t_{i+1})\), we have
Thus
Case 2. \({\gamma }((a,b))\subset V\) and \({\gamma }(\{a,b\})\not \subset V\). Applying Case 1 to \({\gamma }|_{[a+\epsilon ,b-\epsilon ]}\) for sufficiently small \(\epsilon >0\), we get
Case 3. \({\gamma }((a,b))\not \subset V\) and \({\gamma }(\{a,b\})\subset V\). Set
Then \({\gamma }(t_*) \in \partial V\) and \({\gamma }(t^*) \in \partial V\). Write
Note that
By this, and applying Case 2 to \({\gamma }|_{[a,t_*]}\) and \({\gamma }|_{[t^*,b]}\), we obtain
Case 4. \({\gamma }((a,b))\not \subset V\) and \({\gamma }(\{a,b\})\not \subset V\). If \({\gamma }(\{a,b\})\subset \partial V\), then
If \({\gamma }(a)\in V\) and \({\gamma }(b)\in \partial V\), set \(s^*=\min \{s\in [a,b] \ | \ {\gamma }(s) \in \partial V\}\). Obviously \(a<s^*\le b\), and we can find a sequence of \(\epsilon _i>0\) so that \(\epsilon _i\rightarrow 0\) as \(i\rightarrow \infty \) and \({\gamma }(s^*-\epsilon _i)\in V\). Write
Since \({\gamma }(s^*),{\gamma }(b)\in \partial V\), we have
Applying Case 3 to \({\gamma }|_{[a,s^*-\epsilon _i]}\), one has
We therefore have
If \({\gamma }(a)\in \partial V\) and \({\gamma }(b)\in V\), we could prove in a similar way. Thus (5.10) holds and the proof is complete.\(\square \)
The following will be used in Sect. 6. Let \(U\Subset \Omega \) and \(u\in \dot{W}^{1,\infty }_X(U)\cap C^0(\overline{U}) \) satisfying
for some \(0\le \mu <\infty \). Given any subdomain \(V\subset U\), write \(h =u|_{\partial V}\) as the restriction of u in \(\overline{V}\). Since \(d^U_\mu \le d^V_\lambda \) in \(\overline{V}\times \overline{V}\) as given in Lemma 5.3, one has
and hence \(\mu (u|_{\partial V}, \partial V) \le \mu \). Denote by \({\mathcal S}_{u|_{\partial V} ,V}^\mp \) the McShane extension of \(u|_{\partial V} \) in V. Define
Lemma 5.9
Under the assumption (5.12) for some \(0\le \mu <\infty \), the functions \(u^\pm \) defined in (5.13) are continuous in \(\overline{U}\) and satisfy
In particular, \(u^\pm \in \dot{W}^{1,\infty }_X(U) \cap C^0(\overline{U})\) and \( \Vert H(\cdot ,Xu^\pm )\Vert _{L^\infty (U)}\le \mu \).
Proof
We only prove Lemma 5.9 for \(u^+\); the proof of Lemma 5.9 for \(u^-\) is similar. By Lemma 5.6, \({\mathcal S}^+_{h,V}\in C^0(\overline{V})\) and \({\mathcal S}^+_{h,V}=u\) on \(\partial V\). These, together with \(u\in C^0(\overline{U})\) imply that \(u^+\in C^0(\overline{U})\). Moreover, by Corollary 5.1, if \(u^+\) satisfies (5.14), then \(u^+\in \dot{W}^{1,\infty }_X(U)\) and \( \Vert H(\cdot ,Xu^\pm )\Vert _{L^\infty (U)}\le \mu \). Below we prove (5.14) for \(u^+\) via the following 3 cases. By the right continuity of \(\lambda \in [0,\infty )\rightarrow d_\lambda ^U(x,y)\) for any \(x,y\in \overline{U}\), up to considering \( d_{\mu +\epsilon }^U\) for sufficiently small \(\epsilon >0\), we may assume that \(\mu >0\).
Case 1. \(x,y\in \overline{U}\setminus V \). By (5.12) we have
Case 2. \(x \in \overline{V} \) and \(y \in V\) or \(x \in V \) and \(y \in \overline{V}\). Applying Lemma 5.8 with \((U, d_\lambda ^U,V,d^V_\lambda ,g)\) replaced by \(( U,d^U_\mu ,V,d^V_\mu , u|_{\partial V})\), one has
Case 3. \(x\in V\) and \(y\in \overline{U}\setminus V \) or \(x\in \overline{U}\setminus V\) and \(y\in V \). For any \(\epsilon >0\), by Corollary 5.1, there exists a curve \({\gamma }\) joining x, y such that
Let \(z\in {\gamma }\cap \partial V\). Applying Case 2 and Case 1, we have
By the arbitrariness of \(\epsilon >0\), we have \( u^+(y)-u^+(x) \le d^U_\mu (x,y) \) as desired.\(\square \)
6 Proof of Theorem 1.6
In this section, we always suppose that the Hamiltonian H(x, p) enjoys assumptions (H0)-(H3) and further that \(\lambda _H=0\).
Definition 6.1
Let \(U\Subset \Omega \) be a domain and \(g\in C^0(\partial U)\) with \(\mu (g,\partial U)<\infty \).
-
(i)
A minimizer u for \({ \textbf{I}}(g,U)\) is called a local superminimizer for \({ \textbf{I}}(g,U)\) if \(u \ge {\mathcal S}_{u|_{\partial V}; V}^-\) in V for any subdomain \(V\subset U\).
-
(ii)
A minimizer u for \({ \textbf{I}}(g,U)\) is called a local subminimizer for \({ \textbf{I}}(g,U)\) if \(u \le {\mathcal S}_{u|_{\partial V}; V}^+\) in V for any subdomain \(V\subset U\).
The next lemma shows McShane extensions are local super/sub minimizers.
Lemma 6.2
Let \(U\Subset \Omega \) and \(g\in C^0(\partial U)\) with \(\mu (g,\partial U)<\infty \).
-
(i)
For any subdomain \(V\subset U\), we have
$$\begin{aligned} {\mathcal S}_{h^+, V}^- \le {\mathcal S}_{h^+, V}^+ \le \mathcal {S}_{g;U}^+ \text { in }V,\quad \text { where } h^+=\mathcal {S}_{g;U}^+|_{\partial V}. \end{aligned}$$(6.1)In particular, \(\mathcal {S}_{g;U}^+\) is a local superminimizer for \({ \textbf{I}}(g,U)\).
-
(ii)
For any subdomain \(V\subset U\), we have
$$\begin{aligned} \mathcal {S}_{g;U}^-\le {\mathcal S}_{h^-, V}^- \le {\mathcal S}_{h^-, V}^+ \text { in }V,\text { where}\quad h^-=\mathcal {S}_{g;U}^-|_{\partial V}. \end{aligned}$$(6.2)In particular, \(\mathcal {S}_{g;U}^-\) is a local subminimizer for \({ \textbf{I}}(g,U)\).
Proof
We only prove (i); the proof for (ii) is similar. Write \(\mu =\mu (g,\partial U).\) By Lemma 5.7, \(\mathcal {S}_{g;U}^+\) is a minimizer for \(\textbf{I}(g,U)\), \(\mu =\textbf{I}(g,U)=\Vert H(\cdot ,X\mathcal {S}_{g;U}^+)\Vert _{L^\infty (U)} \), and
Fix any subdomain \(V\subset U\). Denote by \({\mathcal S}_{h,V}^\pm \) the McShane extension of \(h^+=\mathcal {S}_{g;U}^+|_{\partial V}\) in V. Note that \({\mathcal S}_{h, V}^-\le {\mathcal S}_{h, V}^+ \) in \(\overline{V}\), that is, the first inequality in (6.1) holds.
Below we show \({\mathcal S}_{h, V}^+\le \mathcal {S}_{g;U}^+\) in V, that is, the second inequality in (6.1). Let \( u^+\) be as in the (5.13) with \(u={\mathcal S}_{g,U}^+\), that is,
By (6.3), we apply Lemma 5.9 to conclude that \(u^+\in \dot{W}^{1,\infty }_X(U) \cap C^0(\overline{U})\) and \( \Vert H(\cdot ,Xu^+)\Vert _{L^\infty (U)}\le \mu \). Note that \(u^+=\mathcal {S}_{g;U}^+\) on \(\partial U\) and hence, by definition of \(\textbf{I}(g, U )\), one has \(\textbf{I}(g, U )\le \Vert H(\cdot ,Xu^+)\Vert _{L^\infty (U)}.\) Recalling \(\mu = \textbf{I}(g, U )\), one obtains that \(\textbf{I}(g, U )= \Vert H(\cdot ,Xu^+)\Vert _{L^\infty (U)},\) that is, \(u^+\) is a minimizer for \(\textbf{I}(g, U )\). By Lemma 5.7(i) again, \(u^+ \le \mathcal {S}_{g;U}^+\) in U. Since \({\mathcal S}_{h, V}^+= u^+\) in V, we conclude that \({\mathcal S}_{h, V}^+\le \mathcal {S}_{g;U}^+\) in V as desired. The proof is complete.\(\square \)
Lemma 6.3
Let \(V \Subset \Omega \) be a domain and \(P=\{x_j\}_{j \in {\mathbb N}}\subset V\) be a dense subset of V. Assume \(u \in C^{0}(\overline{V})\) and \(\{u_j\}_{j \in {\mathbb N}} \subset \dot{W}_X^{1,\infty }(V) \cap C^{0}(\overline{V})\) such that for any \(j \in {\mathbb N}\),
and
Then \(u_j \rightarrow u\) in \(C^0(V)\), and moreover,
Proof
We only need to prove \(u_j \rightarrow u\) in \(C^0(V)\). Note that (6.6) follows from this and Lemma 3.1.
Since \(u \in C^0(\overline{V})\) and \(\overline{V}\) is compact, u is uniform continuous in \(\overline{V}\), that is, for any \(\epsilon >0\), there exists \(h_\epsilon \in (0,\epsilon )\) such that for all
Recalling the assumption (H3), by (6.5) one has
By Lemma 2.7,
Given any \(K\Subset V\), recall from [41] that
It then follows
Given any \(\epsilon >0\), thanks to the density of \(\{x_i\}_{i\in {\mathbb N}}\) in V, one has \(\overline{K}\subset \cup _{x_i\in \overline{K}}B(x_i,h_\epsilon )\). By the compactness of \(\overline{K}\), we have
for some \(i_K\in {\mathbb N}\). For any \(j\ge \max \{i_K,1/\epsilon \}\) and for any \(x\in \overline{K}\), choose \(1\le i\le i_K\) such that \(x_i\in \overline{K}\) and \(|x-x_i|\le h_\epsilon <\epsilon \). Thus \(| u(x_i)-u(x)|\le \epsilon \). By (6.4) we have \(| u_j(x_i)-u(x_i)|\le \frac{1}{j}\le \epsilon \). Thus
This implies that \(u_j\rightarrow u\) in \(C^0(\overline{K})\) as \(j\rightarrow \infty \). The proof is complete.\(\square \)
The following clarifies the relations between absolute minimizers and local super/subminimizers.
Lemma 6.4
Let \(U\Subset \Omega \) and \(g\in C^0(\partial U)\) with \(\mu (g,\partial U)<\infty \). Then a function \(u:\overline{U}\rightarrow {\mathbb R}\) is an absolute minimizer for \({ \textbf{I}}(g,U)\) if and only if it is both a local superminimizer and a local subminimizer for \({ \textbf{I}}(g,U)\).
Proof
If u is an absolute minimizer for \(\textbf{I}(g, U )\), then for every subdomain \(V \subset U\), u is a minimizer for \({ \textbf{I}}(u|_{\partial V},V)\). By Lemma 5.6, \({\mathcal S}_{u|_{\partial V}, V}^- \le u \le {\mathcal S}_{u|_{\partial V}, V}^+\), that is, u is both a local superminimizer and a local subminimizer for \({ \textbf{I}}(g,U)\).
Conversely, suppose that u is both a local superminimizer and a local subminimizer for \({ \textbf{I}}(g,U)\). We need to show that u is absolute minimizer for \(\textbf{I}(g,U)\). It suffices to prove that for any domain \(V\Subset U\), u is a minimizer for \(\textbf{I}(u, V )\), in particular, \(\Vert H(\cdot ,Xu (\cdot ))\Vert _{L^\infty (V)} \le \textbf{I}(u, V)\).
The proof consists of 3 steps.
Step 1. Given any subdomain \(V \subset U\), choose a dense subset \(\{x_j\}_{j \in {\mathbb N}}\) of V. Set \(V_j=V\setminus \{x_i\}_{1\le i\le j}\) and \(V_0=V\). Note that
For each \(j\ge 0\), set
Since \(\overline{V} \subset \overline{U}\), we have \(d_\mu ^U(x,y)\le d_\mu ^V(x,y)\) for all \( x,y\in \overline{V}^*\), we have
and hence \(\mu _0\le \mu \). By Lemma 5.7 (i), \(\textbf{I}(u, V_0)=\mu _0\). In a similar way and by induction, for all \(j\ge 0\), since \(V_{j+1}\subset V_j\), we have
Step 2. We construct a sequence \(\{u_j\}_{j \in {\mathbb N}}\) of functions so that, for each \(j\in {\mathbb N}\),
and
For any \(j \ge 1\), since u is both a local superminimizer and a local subminimizer for \({ \textbf{I}}(g,U)\), by Definition 6.1,
At \(x_{j}\), we have
Define \(u_{j}: \overline{V}_{j-1} =\overline{V} \rightarrow {\mathbb R}\) by
To see (6.10), observe that Lemma 5.7 gives \(u_j\in \dot{W}^{1,\infty }_X(V_j)\cap C^0(\overline{V}) \). Moreover, for any \(x\in \partial V_j\), one has either \(x\in \partial V_{j-1}\) or \(x=x_j\). In the case \(x\in \partial V_{j-1}\), by Lemma 5.7one has
In the case \(x=x_{j}\), if \(a_{j}=b_{j}\), then (6.12) implies
if \(a_{j}<b_{j}\), then
To see (6.11), applying Lemma 5.7(iii) with \(t=\frac{u(x_{j})-a_{j}}{b_{j}-a_{j}}\), we deduce that \(u_{j}\) is a minimizer for \(\textbf{I}(u|_{\partial V_{j-1}},V_{j-1})\), that is,
Step 3. We show that, for all \(j \in {\mathbb N}\),
Note that, by Corollary 5.1 in V, (6.14) yields that \(u_j \in \dot{W}^{1,\infty }_X(V)\) and \(\Vert H(x,Xu_j)\Vert _{L^\infty (V)}\le \mu \). Applying Lemma 6.3 to \(\{u_j\}_{j\in {\mathbb N}}\) and u, we conclude that \(u\in \dot{W}^{1,\infty }_X(V)\) and \(\Vert H(x,Xu )\Vert _{L^\infty (V)}\le \mu \) as desired.
To see (6.14), using (6.11) and Corollary 5.1 in \(V_{j-1}\), we have
Thus (6.9) implies
Thanks to Lemma 5.4 we have \(d^{V_{j-1}}_{\mu }=d_\mu ^V\) in \(V\times V\) and hence
By the continuity of \(u_j\) in \(\overline{V}\) we have (6.14) and hence finish the proof.\(\square \)
Finally, using Perron’s approach, we obtain the existence of absolute minimizers.
Proposition 6.5
Let \(U\Subset \Omega \) and \(g\in C^0(\partial U)\) with \(\mu (g,\partial U)<\infty \). Define
and
Then \(U^\pm _g\) are absolute minimizers for \({ \textbf{I}}(g,U)\).
Proof
We only show that \(U^+_g\) is an absolute minimizer for \({ \textbf{I}}(g,U)\); similarly one can prove that \(U^-_g\) is also an absolute minimizer for \({ \textbf{I}}(g,U)\). Thanks to Lemma 6.4, it suffices to show that \(U^+_g\) is a minimizer for \(\textbf{I}(g,U)\), a local subminimizer for \(\textbf{I}(g,U)\) and a local superminimizer for \(\textbf{I}(g,U)\). Note that \(\textbf{I}(g,U)=\mu (g,\partial U)<\infty . \)
Prove that \(U^+_g\) is a minimizer for \(\textbf{I}(g,U)\). Firstly, since any local subminimizer w for \(\textbf{I}(g,U)\) is a minimizer for \(\textbf{I}(g,U)\). We know
Recalling the assumption (H3), we have \( \Vert |Xw |\Vert _{L^\infty (U)} \le R_{\textbf{I}(g,U)} \) and hence \(w\in {\mathrm {\,Lip}}_{d_{CC}}( \overline{U})\) with \({\mathrm {\,Lip}}_{d_{CC}}(w,\overline{U})\le R_{\textbf{I}(g,U)}\). By a direct calculation, one has
Next, let \(\{x_i\}_{i \in {\mathbb N}}\) be a dense set of U. For any \(i,j\in {\mathbb N}\), by the definition of \(U_g^+\), there exists a local subminimizer \(u_{ij} \) for \(\textbf{I}(g,U)\) such that
Note that, by Definition 6.1, \(u_{ij}\) is also a minimizer for \({ \textbf{I}}(g,U)\).
Moreover, for each \(j\in {\mathbb N}\), write
Lemma 5.7(iii) implies that \(u_j\) is a minimizer for \(\textbf{I}(g,U)\) and hence
For \(1\le i\le j\), from the definition of \(U_g^+\), it follows that
Finally, combining (6.16), (6.17) and (6.18), we are able to apply Lemma 6.3 to \(U_g^+\) and \(\{u_j\}_{j \in {\mathbb N}}\) so to get
Hence
Together with (6.16) yields that \(U^+_g\) is a minimizer for \(\textbf{I}(g,U)\).
Prove that \(U^+_g\) is a local subminimizer for \(\textbf{I}(g,U)\).
We argue by contradiction. Assume on the contrary that \(U^+_g\) is not a local subminimizer for \(\textbf{I}(g,U)\). Then, by definition, there exists a subdomain \(V \subset U\), and some \(x_0\) in V such that
where \({\mathcal S}_{h^+,V}\) is the McShane extension in V of \(h^+=U^+_g|_{\partial V}\). By the definition of \(U^+_g\), there exists a local subminimizer u for \(\textbf{I}(g,U)\) such that
The definition of \(U^+_g\) also gives
Define
Since both u and \({\mathcal S}_{h^+,V}^+\) are continuous, E is an open subset of \(\overline{V}\). Since
by (6.21), we infer that \(u \le {\mathcal S}_{h^+,V}^+ \text{ on } \partial V\) and hence \(E \subset V.\) Obviously, \(x_0\in E\). Denote by \(E_0\) the component of E containing \(x_0\).
Recalling that u is a local subminimizer for \(\textbf{I}(g,U)\), by Definition 6.1, we have \(u \le {\mathcal S}_{u|_{\partial E_0}, E_0}^+\) in E. Since \(x_0 \in E_0\), we have \( {\mathcal S}_{u|_{\partial E_0}, E_0}^+(x_0) \ge u(x_0 ),\) which, combined with (6.20), gives
On the other hand, we are able to apply Lemma 6.2 with \((U,V,g, h=\mathcal {S}_{g;U}^+|_{\partial V} )\) therein replaced by \((V,E_0, h^+, h= {\mathcal S}^+_{h^+,V}| _{E_0})\) here and then obtain
Since \(u|_{E_0}= {\mathcal S}^+_{h^+,V}| _{E_0}=h\), at \(x_0 \in E\), we arrive at
Note that (6.22) contradicts with (6.23) as desired.
Prove that \(U^+_g\) is a local superminimizer for \(\textbf{I}(g,U)\). By definition, it suffices to prove that, for any given subdomain \(V \subset U\), we have \({\mathcal S}_{h^+, V}^-\le U^+_g\) in V, where we write \(h^+=U^+_g|_{\partial V}\).
To this end, define \(u^+\) as in (5.13) with u therein replaced by \(U^+_g\), that is,
Then \(u^+\) is a minimizer for \(\textbf{I}(g,U)\). Indeed, since \(U^+_g\) is a minimizer for \(\textbf{I}(g,U)\), we know that \(U^+_g\) satisfies (5.12) with \(\mu =\textbf{I}(g,U)\) therein. This allows us to apply Lemma 5.9 with \(u= U^+_g\) therein and then conclude that \(u^+\in \dot{W}^{1,\infty }_X(U)\cap C^0(U)\), \(u^+=U^+_g =g\) on \(\partial U\), and \(\Vert H(x,Xu^+ )\Vert \le \textbf{I}(g,U) \). Therefore, by definition of \(\textbf{I}(g,U) \), \(\Vert H(x,Xu^+ )\Vert = \textbf{I}(g,U) \), and hence \(u^+\) is a minimizer for \(\textbf{I}(g,U)\).
We further claim that
Assume that this claim holds. Choosing \(u^+\) as a test function in the definition of \(U^+_g\) in (6.15), we know that
and in particular \(U^+_g \ge {\mathcal S}_{h^+, V}^- \) in V as desired.
Proof the claim (6.25).
To get (6.25), by Definition 6.1, we still need to show for any subdomain \(B \subset U\),
To prove (6.26), we argue by contradiction. Assume that (6.26) is not correct, that is,
Up to considering some connected component of W, we may assume that W is connected. Note that
Consider the set
By continuity, both of W and D are open.
Below, we consider two cases: \(D=\emptyset \) and \(D\ne \emptyset \).
Case \(D=\emptyset \). If D is empty, then we always have \(U^+_g \le {\mathcal S}_{u^+|_{\partial B},B}^+\) in B. Thus
Since \(U^+_g=u^+\in \overline{U}\setminus V\), this implies
Since \(u^+= {\mathcal S}_{h^+, V}^-\) in \(\overline{V}\) and \(W \subset V\) gives \( \partial W \subset \overline{V}\), we have
Moreover, by Lemma 6.2, we know that \( {\mathcal S}_{h^+, V}^-\) with \(h^+=U^+_g|_{\partial V}\) is a local subminimizer for \(\textbf{I}(U^+_g,V)\). By the definition of local subminimizer, and by \(W\subset V\), we have
where we recall \({\mathcal S}_{h^+, V}^-|_{\partial W}=u^+|_{\partial W}\) from (6.31).
Applying Lemma 6.2 with \((U,V,g,h^+)\) therein replaced by \((B,W, u^+|_{\partial B}, {\mathcal S}_{u^+|_{\partial B},B}^+|_{\partial W})\) here, recalling \({\mathcal S}_{u^+|_{\partial B},B}^+|_{\partial W}=u^+|_{\partial W}\) from (6.28), we obtain
Combing (6.32) and (6.33), by \(W\subset V\), one arrives at
which contradicts with (6.27).
Case \(D\ne \emptyset \). Up to considering some connected component of D, we may assume that D is connected. By the definition of D in (6.29), we infer that
Since \(U^+_g\) is a local subminimizer for \(\textbf{I}(g,U)\) as proved above, we know
Applying Lemma 6.2 with \((U,V,g,h^+)\) therein replaced by \((B,D, u^+|_{\partial B}, {\mathcal S}_{u^+|_{\partial B},B}^+|_{\partial D}) \), recalling \({\mathcal S}_{u^+|_{\partial B},B}^+|_{\partial D}=U^+_g |_{\partial D}\) from (6.34), we obtain
Combining (6.35) and (6.36), we deduce
Recalling (6.29), this contradicts with \(D\ne \emptyset \). The proof is complete. \(\square \)
Theorem 1.6 is now a direct consequence of the above series of results.
Proof of Theorem 1.6
Let \(g\in {\mathrm {\,Lip}}_{d_{CC}} (\partial U)\). It suffices to show that \(\mu (g,\partial U)<\infty \), which allows us to use Proposition 6.5 and then conclude the desired absolute minimizer \(U_g^+ \) therein.
Taking \(0<\lambda <\infty \) such that \(R_\lambda ' \ge {\mathrm {\,Lip}}_{d_{CC}}(g,\partial U)\), we have
From Lemma 2.9 (ii), that is, \(R'_\lambda d_{CC} \le d^U_\lambda ,\) it follows that
and hence that
The proof is complete.\(\square \)
7 Further discussion
Note that Rademacher type Theorem 1.3 is a cornerstone when showing the existence of absolute minimizers. Indeed, Champion and Pascale [14] and Guo-Xiang-Yang [26] established partial results similar to Theorem 1.3 for a special class of Hamiltonians considered in this paper to show the existence of absolute minimizers. However, their method seems to be invalid for more general Hamiltonians considered in this paper. We briefly explain the reason below.
Remark 7.1
Champion and Pascale [14] showed the McShane extension is a minimizer for H when H is lower semi-continuous on \(U \times {{{\mathbb R}}^n}\). In fact, they defined another intrinsic distance induced by H(x, p). For every \(\lambda \ge 0\),
where \(H_\lambda (x)\) is the sub-level set at x, namely, \(H_\lambda (x)=\{p \in {\mathbb R}^m \mid H(x,p) \le \lambda \}\).
For \(0\le a<b\le +\infty \), let \({\gamma }:[a,\,b]\rightarrow \overline{U}\) be a Lipschitz curve, that is, there exists a constant \(C>0\) such that \(|{\gamma }(s)-{\gamma }(t)|\le C|s-t|\) whenever \(s,t\in [a,b]\). The \(L_\lambda \)-length of \({\gamma }\) is defined by
which is nonnegative, since \(L_\lambda (x,q)\ge 0\) for any \(x\in \overline{U}\) and \(q\in {{{\mathbb R}}^n}\). For a pair of points \(x,\,y\in \overline{U}\), the \(\overline{d}_{\lambda }\)-distance from x to y is defined by
Then, they prove two intrinsic pseudo-distance are equal, that is
Thanks to the definition of \(\overline{d}_{\lambda }\), they can justify (i)\(\Leftrightarrow \)(ii) in Rademacher type Theorem 1.3.
However, when asserting (7.1), we will meet obstacles in generalizing [14] Proposition A.2 since we are faced with measurable H.
On the other hand, Guo-Xiang-Yang [26] provided another method to identify a weak version of (7.1) for measurable Finsler metrics H, that is
Here \(\widetilde{d}_\lambda \) induced by measurable Finsler metrics H is defined in the following way.
where the supremum is taken over all subsets N of \( \overline{U}\) such that \(|N| = 0\) and \(\Gamma _N(a,b,x,y, U)\) denotes the set of all Lipschitz continuous curves \({\gamma }\) in \( \overline{U}\) with end points x and y such that \(\mathcal H^1(N \cap {\gamma }) = 0\) with \(\mathcal H^1\) being the one dimensional Hausdorff measure.
In fact, (7.2), combined with the method in [14] will be sufficient for validating (i)\(\Leftrightarrow \)(ii) in Rademacher type Theorem 1.3. Unfortunately, since we are coping with Hörmander vector field, a barrier arises when modifying their proofs. Indeed, their method uses a result by [16] that every x-measurable Hamiltonian H can be approximated by a sequence of smooth Hamiltonians \(\{H_n\}_n\) such that two intrinsic distances \(\widetilde{d}_\lambda ^{H_n}\) and \(d_\lambda ^{H_n}\) induced by \(H_n\) by means of (7.3) and (1.4) satisfy \( \lim _{n \rightarrow \infty }d_\lambda ^{H_n} = d_\lambda \) and \( \limsup _{n \rightarrow \infty }\widetilde{d}_\lambda ^{H_n} \le \widetilde{d}_\lambda \) uniformly on \( U\times U\) respectively. The process of the proof of [16] is based on a \(C^1\) Lusin approximation property for curves. Namely, given a Lipschitz curve \({\gamma }: \ [0,1] \rightarrow U\) joining x and y, for any \(\epsilon >0\), there exists a \(C^1\) curve \(\widetilde{{\gamma }}: \ [0,1] \rightarrow U\) with the same endpoints such that
where \(\mathcal {L}^1\) denotes the one dimensional Lebesgue measure. Besides,
for some constant c depending only on n. Although this version of \(C^1\) Lusin approximation property holds for horizontal curves in Heisenberg groups ([43]) and step 2 Carnot groups ([39]), it fails for some horizontal curve in Engel group ([43]).
In summary, it is difficult to generalize the properties of the pseudo metric \(\widetilde{d}_\lambda \) not only from Euclidean space to the case of Hörmander vector fields but also from lower-semicontinuous H(x, p) to measurable H(x, p). Hence we would like to pose the following open problem.
Problem 7.2
Under the assumptions (H0)-(H3), does (7.2) holds?
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Communicated by Andrea Mondino.
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The first author is supported by the Academy of Finland via the projects: Quantitative rectifiability in Euclidean and non-Euclidean spaces, Grant No. 314172, and Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups, Grant No. 328846. The second author is supported by the National Natural Science Foundation of China (No. 12025102 & No. 11871088) and by the Fundamental Research Funds for the Central Universities. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Appendix A: Rademacher’s theorem in Euclidean domains—revisit
Appendix A: Rademacher’s theorem in Euclidean domains—revisit
In this appendix, we state some consequence of Rademacher’s theorem (Theorem 1.1) for Sobolev and Lipschitz classes, see Lemma 3 and Lemma 6 below. They were well-known in the literature and also partially motivated our Theorem 1.3 and Corollary 1.5. For reader’s convenience, we give the details.
Recall that \(\Omega \subset {{{\mathbb R}}^n}\) is always a domain. The homogeneous Sobolev space \(\dot{W}^{1,\infty }(\Omega )\) is the collection of all functions \(u\in L^1_{\mathrm {\,loc\,}}(\Omega )\) with its distributional derivative \(\nabla u=(\frac{\partial u}{\partial x_i})_{1\le i\le n}\in L^\infty (\Omega )\). We equip \(\dot{W}^{1,\infty }(\Omega )\) with the semi-norm
Write \(\dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\) as the collection of all functions u in \(\Omega \) so that \(u\in \dot{W}^{1,\infty }(V)\) whenever \(V\Subset \Omega \). Here and below, \(V\Subset \Omega \) means that V is bounded domain with \(\overline{V}\subset \Omega \). On the other hand, denote by \({\mathrm {\,Lip}}(\Omega )\) the collection of all Lipschitz functions u in \(\Omega \), that is, all functions u satisfying (1.1). We equip \({\mathrm {\,Lip}}(\Omega )\) with the semi-norm
Denote by \({\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\) the collection of all functions u in \(\Omega \) so that \(u\in {\mathrm {\,Lip}}(V)\) for any \(V\Subset \Omega \). Moreover, denote by \({\mathrm {\,Lip}}^*(\Omega )\) the collection of all functions u in \(\Omega \)
with
Obviously, \( {\mathrm {\,Lip}}(\Omega )\subset {\mathrm {\,Lip}}^*(\Omega )\) with the seminorm bound \({\mathrm {\,Lip}}^*(u,\Omega )\le {\mathrm {\,Lip}}(u,\Omega ).\) Next, we have the following relation.
Lemma 1
We have \( {\mathrm {\,Lip}}^*(\Omega )\subset {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\). For any convex subdomain \(V\subset \Omega \) and \(u\in {\mathrm {\,Lip}}^*(\Omega )\), we have
Proof
Let \(u\in {\mathrm {\,Lip}}^*(\Omega )\). To see \( u\in {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\), it suffices to prove \(u\in {\mathrm {\,Lip}}(B)\) for any ball \(B\Subset \Omega \). Given any \(x,y\in B\), denote by \({\gamma }(t)=x+t(y-x)\). Since \(A_{x,y}=\sup _{t\in [0,1]} {\mathrm {\,Lip}}u({\gamma }(t))<\infty \), for each \(t\in [0,1]\) we can find \(r_t>0\) such that \(|u({\gamma }(s))-u({\gamma }(t))|\le A_{x,y}|{\gamma }(s)-{\gamma }(t)|=A_{x,y}|s-t||x-y| \) whenever \(|s-t|\le r_t\) and \(s \in [0,1]\). Since \([0,1]\subset \cup _{t\in [0,1]}(t-r_t,t+r_t)\) we can find an increasing sequence \(t_i\in [0,1]\) with \(t_0 = 0\) and \(t_N=1\) such that \([0,1]\subset \cup _{i=1}^N(t_i-\frac{1}{2}r_{t_i},t_i+\frac{1}{2} r_{t_i})\). Write \(x_i={\gamma }(t_i)\) for \(i =0 , \cdots , N\). We have
Noticing that \(A_{x,y} \le {\mathrm {\,Lip}}^*(u,B) \le {\mathrm {\,Lip}}^*(u,\Omega ) < \infty \) for all \(x,y \in B\), we deduce that \(u \in {\mathrm {\,Lip}}(B)\) and hence \(u \in {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\).
If \(V \Subset \Omega \) is convex, then for any \(x,y \in \Omega \), the line-segment joining x and y lies in V. Hence similar to the above discussion, we have
and \(A_{x,y} \le \Vert u\Vert _{{\mathrm {\,Lip}}^*(V)}< \infty \) for all \(x,y \in V\). Therefore, (2) holds and the proof is complete.\(\square \)
On the other hand, functions \(\dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\) admit continuous representatives.
Lemma 2
(i) Each \(u\in \dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\) admits a unique continuous representative \(\widetilde{u}\), that is, \(\widetilde{u}\in \dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\) with \(\widetilde{u}=u\) almost everywhere in \(\Omega \). Moreover, \(\widetilde{u}\in {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\), and for any convex subdomain \(V\subset \Omega \), we have
(ii) Each \(u\in \dot{W}^{1,\infty } (\Omega )\) admits a unique continuous representative \(\widetilde{u}\), that is, \(\widetilde{u}\in \dot{W}^{1,\infty } (\Omega )\) with \(\widetilde{u}=u\) almost everywhere in \(\Omega \). Moreover, \(\widetilde{u}\in {\mathrm {\,Lip}}^*(\Omega )\) with \({\mathrm {\,Lip}}^*(\widetilde{u},\Omega )\le \Vert u\Vert _{\dot{W}^{1,\infty }(\Omega )}\).
Proof
Since (ii) can be shown in a similar way as (i), we only prove (i). Given any convex domain \(V\Subset \Omega \), for any pair x, y of Lebesgue points of u, we have
where \(\eta _\delta \) is the standard mollifier in \({\mathbb R}^n\) with its support \(\textrm{spt}\eta _\delta \subset B(0,\delta )\) and \(t_\delta \in [0,1]\). Also, since for any \(z\in V\),
we deduce that for any pair x, y of Lebesgue points of u,
If \(z \in V\) is not a Lebesgue point of u, let \(\{z_i\}_{i \in {\mathbb N}} \subset V\) be a sequence of Lebesgue points of u converging to z. We have
which implies \(\{u(z_i)\}_{i \in {\mathbb N}} \subset V\) is a Cauchy sequence. Since \(\Vert u\Vert _{L^\infty (V)} < \infty \), we know \(\{u(z_i)\}_{i \in {\mathbb N}}\) has a limit in \({\mathbb R}\) independent of the choice of the sequence \(\{u(z_i)\}_{i \in {\mathbb N}}\). Define \(\widetilde{u}(z):= u(z) \) if \(z \in V\) is a Lebesgue point of u and \(\widetilde{u}(z)= \lim _{i \rightarrow \infty } u(z_i)\) if \(z \in V\) is not a Lebesgue point of u where \(\{z_i\}_{i \in {\mathbb N}} \subset V\) is a sequence of Lebesgue points of u converging to z. We know \(\widetilde{u} : V \rightarrow {\mathbb R}\) is well-defined and moreover,
Thus \(\widetilde{u}\in {\mathrm {\,Lip}}(V)\) with \(\sup _{x\in V}{\mathrm {\,Lip}}\widetilde{u}(x)\le {\mathrm {\,Lip}}(\widetilde{u},V)\le \Vert \nabla u\Vert _{L^\infty (V)}\). In particular, \(\widetilde{u}\) is continuous, which shows (i).\(\square \)
Thanks to lemma 2, below for any function \(u\in \dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\) or \(u\in \dot{W}^{1,\infty }(\Omega )\), up to considering its continuous representative \(\widetilde{u}\), we may assume that u is continuous. Under this assumption, Lemma 2 further gives \(\dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\subset {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\), and \(\dot{W}^{1,\infty } (\Omega )\subset {\mathrm {\,Lip}}^*(\Omega )\) with a norm bound \({\mathrm {\,Lip}}^*(u,\Omega )\le \Vert u\Vert _{\dot{W}^{1,\infty }(\Omega )}\). Rademacher’s theorem (Theorem 1.1) tells that their converse are also true. Indeed, we have the following.
Lemma 3
(i) We have \( \dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )={\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\) and \({\mathrm {\,Lip}}( \Omega )\subset \dot{W}^{1,\infty } (\Omega )= {\mathrm {\,Lip}}^*(\Omega )\) with \({\mathrm {\,Lip}}^*(u,\Omega )= \Vert u\Vert _{\dot{W}^{1,\infty }(\Omega )}\le {\mathrm {\,Lip}}(u,\Omega )\).
(ii) If \(\Omega \) is convex, then \({\mathrm {\,Lip}}( \Omega )= \dot{W}^{1,\infty } (\Omega )= {\mathrm {\,Lip}}^*(\Omega )\) with \({\mathrm {\,Lip}}^*(u,\Omega )= \Vert u\Vert _{\dot{W}^{1,\infty }(\Omega )}= {\mathrm {\,Lip}}(u,\Omega )\).
Proof
(i) If \(u\in {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\), applying Rademacher’s theorem (Theorem 1.1) to all subdomains \(V\Subset \Omega \), one has \(u\in \dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\) and \( |\nabla u(x)|={\mathrm {\,Lip}}u(x)\) for almost all \(x\in \Omega \) (whenever u is differentiable at x). Hence \({\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega ) \subset W^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\). Combining Lemma 2(i), we know \({\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega ) = W^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\).
If \(u\in {\mathrm {\,Lip}}^*(\Omega )\), that is, \({\mathrm {\,Lip}}^*(u,\Omega ) = \sup _{x\in \Omega }{\mathrm {\,Lip}}u(x)<\infty \). We have \(u\in {\mathrm {\,Lip}}_{{\mathrm {\,loc\,}}}(\Omega )\) and hence \(u\in \dot{W}^{1,\infty }_{\mathrm {\,loc\,}}(\Omega )\) and \( |\nabla u(x)|= {\mathrm {\,Lip}}u(x) \le {\mathrm {\,Lip}}^*(u,\Omega )<\infty \) for almost all \(x\in \Omega \). Thus \(u\in \dot{W}^{1,\infty }(\Omega )\).
By definition, it is obvious that \({\mathrm {\,Lip}}( \Omega )\subset {\mathrm {\,Lip}}^*(\Omega )\). Hence Lemma 3 (i) holds.
(ii) By Lemma 3 (i), we only need to show \({\mathrm {\,Lip}}^*(\Omega ) \subset {\mathrm {\,Lip}}( \Omega )\). Applying Lemma 1 with \(V=\Omega \) therein, (2) becomes
Taking supremum among all \(x,y \in \Omega \) in the left hand side of the above inequality, we arrive at
which gives the desired result.\(\square \)
Remark 4
(i) Lemma 1 and Lemma 3 fail if we relax \( \sup _{x\in \Omega }{\mathrm {\,Lip}}u(x)\) in the definition (1) to be \(\Vert {\mathrm {\,Lip}}u\Vert _{L^\infty (\Omega )}=\mathrm {\,esssup\,}_{x\in \Omega }{\mathrm {\,Lip}}u(x)\). This is witted by the standard Cantor function w in [0, 1]. Denote by E the standard Cantor set. It is well-known that w is continuous but not absolute continuous in [0, 1]. Since Lipschitz functions are always absolutely continuous, we know that w is neither Lipschitz nor locally Lipschitz in \(\Omega =(0,1)\), and hence \(w\notin {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\). On the other hand, observe that \(\Omega \setminus E\) consists of a sequence of open intervals which mutually disjoint, and w is a constant in each such intervals and hence \({\mathrm {\,Lip}}w(x)=0 \) therein. So we know that \({\mathrm {\,Lip}}w(x)=0 \) in \(\Omega \setminus E\). Since \(|E|=0\), we have \(\Vert {\mathrm {\,Lip}}u\Vert _{L^\infty (\Omega )}=0\).
(ii) In general, if \(\Omega \) is not convex, one cannot expect that \( \dot{W}^{1,\infty } (\Omega )\subset {\mathrm {\,Lip}}(\Omega )\) with a norm bound. Indeed, consider the planar domain
Indeed, in the polar coordinate \((r,\theta )\), let \(w: U \rightarrow {\mathbb R}\) be
One can show that \(w\in \dot{W}^{1,\infty }(U)\) so that \(w(x_1,x_2)< \pi /3\) when \(1/2\le x_1<1\) and \(0<x_2<1/10\), and \(w(x_1,x_2)> 5\pi /6\) when \(1/2\le x_1<1\) and \(-1/10<x_2<0\). One has \({\mathrm {\,Lip}}(w,\Omega )=\infty \) and hence \(w\notin {\mathrm {\,Lip}}(\Omega )\).
The example in Remark 4 (ii) also indicates that the Euclidean distance does not match the geometry of domains and hence \({\mathrm {\,Lip}}(\Omega )\) defined via Euclidean distance is not the prefect one to understand \( \dot{W}^{1,\infty } (\Omega )\).
Instead of Euclidean distance, for any domain \(\Omega \), we consider the intrinsic distance
where \(\ell ({\gamma }):=\int _0^1|\dot{\gamma }(t)|\,dt\) is the Euclidean length. We have the dual formula.
Lemma 5
(i) For any \(x,y\in \Omega \),
(ii) If \(x,y\in \Omega \) with \(|x-y|\le {\mathrm {\,dist\,}}(x,\partial \Omega )\), then \(d^\Omega _E(x,y)=|x-y|\).
(iii) If \(\Omega \) is convex, then \(d^\Omega _E(x,y)=|x-y|\) for all \(x,y\in \Omega \).
Proof
(i) Set
Notice that \(d^\Omega _E(x, \cdot ) \in \textrm{Lip}^*(\Omega ) = \dot{W}^{1,\infty }(\Omega )\) (Lemma 3 (i)) and \(\Vert \nabla d^\Omega _E(x, \cdot )\Vert _{L^\infty (\Omega )}\le 1\) for all \(x \in \Omega \). Hence letting \(d^\Omega _E(x, \cdot )\) be the test function in (6), we see
To see the contrary, fix \(x,y \in \Omega \). Let \(\{u_i\}_{i \in {\mathbb N}}\) be a sequence of test functions in (6) such that
Let \({\gamma }:[0,1] \rightarrow \Omega \) be an arbitrary absolute continuous curve joining x and y. Then there exists a domain \(U \Subset \Omega \) with \({\gamma }\subset U\). Let \(\{\eta _\delta \}_{\delta >0}\) be the standard mollifiers in \({\mathbb R}^n\). For each \(i \in {\mathbb N}\), we know \(u_i *\eta _\delta \in C^\infty (\Omega )\) and \(\Vert \nabla (u_i *\eta _\delta )\Vert _{L^\infty (U)}\le \Vert \nabla u_i \Vert _{L^\infty (U)} \le 1\). Then we have
Finally, taking infimum among all absolute continuous curves joining x and y in the above inequality, we conclude
(ii) If \(|x-y|\le {\mathrm {\,dist\,}}(x,\partial \Omega )\), then the line-segment \({\gamma }\) joining x and y is contained in \(\Omega \). Letting \({\gamma }\) be the absolute continuous curve in (4),
(iii) If \(\Omega \) is convex, for all \(x,y\in \Omega \), since the line-segment joining them is contained in \(\Omega \), similarly to (ii), we have \(d^\Omega _E(x,y)=|x-y|\). The proof is complete.\(\square \)
Note that if \(\Omega \) is not convex, one cannot expect \(d^\Omega _E(x,y)=|x-y|\) for all \(x,y\in \Omega \). Indeed, if \(\Omega \) is given by the domain U as in (3), for points \((1/2,\epsilon )\) and \((1/2,-\epsilon )\) with \(\epsilon \in (0,1/10)\), the Euclidean distance between them is \(2\epsilon \). However, note that any curve \({\gamma }:[0,1]\rightarrow \Omega \) joining them must have intersection with \((-1,0)\times \{0\}\), which is call z. One then deduce that
Thus the intrinsic distance between \((1/2,\epsilon )\) and \((1/2,-\epsilon )\) is always larger than or equals to \(1+2\epsilon \).
With in Lemma 5, we show that the Lipschitz spaces defined via the intrinsic distance perfectly match with the Sobolev space \( \dot{W}^{1,\infty }(\Omega )\), see Lemma 6 below. Denote by \({\mathrm {\,Lip}}_{d^\Omega _E}(\Omega ) \) the collection of all Lipschitz functions u in \(\Omega \) with respect to \(d^\Omega _E\), that is,
We also denote by \({\mathrm {\,Lip}}^*_{d^\Omega _E}(\Omega )\) the collection of all functions u in \(\Omega \)
with
Lemma 6
We have \( {\mathrm {\,Lip}}_{d^\Omega _E}(\Omega )= \dot{W}^{1,\infty } (\Omega )= {\mathrm {\,Lip}}^*(\Omega )\) and
Proof
Recall that Lemma 6 gives \(d^\Omega _E(x,y)=|x-y|\) whenever \(|x-y|\le {\mathrm {\,dist\,}}(x,\partial \Omega )\). One then has \({\mathrm {\,Lip}}_{d^\Omega _E}(\Omega ) \subset {\mathrm {\,Lip}}_{\mathrm {\,loc\,}}(\Omega )\), and moreover, \({\mathrm {\,Lip}}u(x)= {\mathrm {\,Lip}}_{d^\Omega _E} u(x)\) for all \(x\in \Omega \), which gives \({\mathrm {\,Lip}}^*_{d^\Omega _E}(\Omega )={\mathrm {\,Lip}}^*(\Omega )\).
Next, we show \(\dot{W}^{1,\infty } (\Omega ) \subset {\mathrm {\,Lip}}_{d^\Omega _E}(\Omega )\) and \( {\mathrm {\,Lip}}_{d^\Omega _E} (u,\Omega ) \le \Vert \nabla u\Vert _{L^\infty (\Omega )}\). Let \(u \in \dot{W}^{1,\infty } (\Omega )\). Then \(\Vert \nabla u\Vert _{L^\infty (\Omega )} =:\lambda < \infty .\) If \( \lambda >0\), then \(\lambda ^{-1}u \in \dot{W}^{1,\infty } (\Omega )\) and \(\Vert \nabla (\lambda ^{-1} u )\Vert _{L^\infty (\Omega )} =1\). Hence \(\lambda ^{-1}u\) could be the test function in (5), which implies
or equivalently,
Therefore, \(u \in {\mathrm {\,Lip}}_{d^\Omega _E}(\Omega )\) and \( {\mathrm {\,Lip}}_{d^\Omega _E} (u,\Omega ) \le \Vert \nabla u\Vert _{L^\infty (\Omega )}\). If \(\lambda =0\), then similar as the above discussion, we have for any \(\lambda '>0\)
Therefore, \(u \in {\mathrm {\,Lip}}_{d^\Omega _E}(\Omega )\) and \( {\mathrm {\,Lip}}_{d^\Omega _E} (u,\Omega ) \le \lambda '\) for any \(\lambda '>0\). Hence \( {\mathrm {\,Lip}}_{d^\Omega _E} (u,\Omega ) =0 = \Vert \nabla u\Vert _{L^\infty (\Omega )}\).
Moreover, we show \( {\mathrm {\,Lip}}_{d^\Omega _E}(\Omega ) \subset {\mathrm {\,Lip}}^*(\Omega )\) and \( {\mathrm {\,Lip}}^*(u,\Omega ) \le {\mathrm {\,Lip}}_{d^\Omega _E} (u,\Omega )\). Let \(u \in {\mathrm {\,Lip}}_{d^\Omega _E}(\Omega )\). Then \({\mathrm {\,Lip}}_{d^\Omega _E} (u,\Omega )<\infty \). Since \({\mathrm {\,Lip}}^*_{d^\Omega _E}(u,\Omega ) \le {\mathrm {\,Lip}}_{d^\Omega _E} (u,\Omega )\) and \({\mathrm {\,Lip}}^*_{d^\Omega _E}(u,\Omega ) = {\mathrm {\,Lip}}(u,\Omega )\), we arrive at
Therefore, \(u \in {\mathrm {\,Lip}}^*(\Omega )\).
Finally, recalling that \(\dot{W}^{1,\infty } (\Omega ) = {\mathrm {\,Lip}}^*(\Omega )\) and \(\Vert \nabla u\Vert _{L^\infty (\Omega )} = {\mathrm {\,Lip}}(u,\Omega ) \) in Lemma 3, we finish the proof.\(\square \)
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Liu, J., Zhou, Y. A Rademacher type theorem for Hamiltonians H(x, p) and an application to absolute minimizers. Calc. Var. 62, 144 (2023). https://doi.org/10.1007/s00526-023-02484-9
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DOI: https://doi.org/10.1007/s00526-023-02484-9