Abstract
In this paper, we study the Cauchy–Dirichlet problem
where \(\Omega \subset \mathbb {R}^n\) is a convex and bounded domain, \(f:[0,T]\times {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) is \(L^1\)-integrable in time and convex in the second variable. Assuming that the initial and boundary datum \(u_o:{\overline{\Omega }}\rightarrow {\mathbb {R}}\) satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.
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1 Introduction and results
It follows from classical theory [15, 17, 18, 26, 29] (see also [14, Chapter 1]) that any variational functional \(F:W^{1,\infty }(\Omega ) \rightarrow {\mathbb {R}}\) of the form
where \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) is convex and \(\Omega \subset {\mathbb {R}}^n\) is a convex domain, admits a unique Lipschitz continuous minimizer in the class \(\{v \in W^{1,\infty } (\Omega ): v=v_o \text { on } \partial \Omega \}\) provided that the boundary datum \(v_o\) satisfies the bounded slope condition (see Definition 2.1). Modern elliptic results involving one-sided bounded slope conditions or more general integrands include for example [2,3,4, 10, 13, 22,23,24].
Surprisingly, while Hardt and Zhou [16, Chapter 4] used the bounded slope condition in a regularity argument in a time-dependent setting involving functionals with linear growth, an evolutionary analogue of the above stationary theorem was established only rather recently by Bögelein, Duzaar, Marcellini and Signoriello [7]. They considered the Cauchy–Dirichlet problem
where \(\Omega _T:= \Omega \times (0,T)\) with \(\Omega \subset \mathbb {R}^n\) and \(T \in (0,\infty ]\) denotes a space-time cylinder and \(\partial _\mathcal {P} \Omega _T:= \partial \Omega \times (0,T) \cup ({{\overline{\Omega }}} \times \{0\})\) its parabolic boundary. Given a Lipschitz continuous initial and boundary datum \(u_o\) that satisfies the bounded slope condition, in [7] it was proven that the above problem admits a unique variational solution that is globally Lipschitz continuous with respect to the spatial variables. Moreover, if the integrand f fulfills an additional p-coercivity condition with some \(p>1\), Bögelein and Stanin [8] obtained the local Lipschitz continuity of variational solutions in space and time under the assumption that \(u_o\) is convex and Lipschitz continuous. Further, global continuity of u was proven in the case that \(\Omega \) is uniformly convex.
For the same class of integrands and merely convex domains \(\Omega \), Stanin [30] showed that variational solutions are still globally Hölder continuous even if the convexity assumption on \(u_o\) is dropped. Equations with lower-order terms were considered by Rainer, Siltakoski and Stanin [27] who extended a stationary Haar-Rado type theorem by Mariconda and Treu [24] to the parabolic problem
where f is convex and p-coercive with some \(p>1\) and the lower-order term g satisfies a technical condition, in particular convexity with respect to u. As a corollary, the authors in [27] obtained the global Lipschitz continuity with respect to the spatial variables of variational solutions under the classical two-sided bounded slope condition provided that \(f \in C^2\) is uniformly convex in a suitable sense.
Existence and regularity of solutions under general growth conditions, such as the so called \(p-q\)-growth conditions, have been recently considered by many authors, see for example [21, 25] and the references therein. We emphasize that in the present manuscript, because of the bounded slope condition, no special growth conditions are imposed on the elliptic part of the operator.
The objective of the present paper is to extend the result of [7] to include time-dependent integrands. In order to focus on the novelty and to include integrands f with linear growth, we consider the classical bounded slope condition and avoid lower-order terms. We are concerned with parabolic partial differential equations of the form
where \(\Omega \subset {\mathbb {R}}^n\) is a convex and bounded domain and \(T \in (0,\infty ]\). The integrand \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) is assumed to be a Carathéodory function that satisfies the following assumptions:
In particular, for any \(L>0\) and \(\tau \in (0, T] \cap \mathbb {R}\) the map \(t \mapsto \max _{|\xi | \le L} |f(t,\xi )|\) belongs to \(L^1(0, \tau )\) (see Sect. 2.3 below). Therefore, for any \(\tau \in (0, T] \cap \mathbb {R}\) and \(V \in L^\infty (\Omega _T,\mathbb {R}^n)\) we have that
We emphasize that \(t \mapsto f(t, \xi )\) is neither assumed to be continuous nor weakly differentiable.
Examples of admissible integrands are functionals with linear growth such as the area integrand \(f(\xi ) = \sqrt{1 + |\xi |^2}\), integrands with exponential growth like \(f(\xi ) = \exp (|\xi |^2)\), Orlicz type functionals such as \(f(\xi ) = |\xi | \log (1 + |\xi |)\) and time-dependent combinations thereof like \(f(t,\xi ) = \chi _{[0,t_o]} f_1(\xi ) + \chi _{(t_o, T]} f_2(\xi )\) or more general \(f(t,\xi ) = \sum _{i=1}^m a_i(t) f_i(\xi )\) for functions \(a_i \in L^1(0,T)\), \(i=1,\ldots ,m\).
In the present paper, we define variational solutions in the same way as in [5]. This notion of solution, inspired by Lichnewsky and Temam [20], was introduced by Bousquet [2, 3] in the time-independent setting. We consider the following class of functions that are Lipschitz continuous in space
Further, we denote the subclass related to time-independent boundary values \(u_o \in W^{1, \infty } (\Omega )\) by
Definition 1.1
(Variational solutions) Assume that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies (1.2) and consider a boundary datum \(u_o \in W^{1,\infty }(\Omega )\). In the case \(T \in (0,\infty )\) a map \(u \in K^\infty _{u_o}(\Omega _T)\) is called a variational solution to the Cauchy–Dirichlet problem associated with (1.1) and \(u_o\) in \(\Omega _T\) if and only if the variational inequality
holds true for any comparison map \(v \in K^\infty _{u_o}(\Omega _T)\) with \(\partial _t v \in L^2(\Omega _T)\). If \(T=\infty \) and \(u \in K^\infty _{u_o}(\Omega _\infty )\) is a variational solution in \(\Omega _\tau \) for any \(\tau \in (0,\infty )\), u is called a global variational solution or variational solution in \(\Omega _\infty \) to the Cauchy–Dirichlet problem associated with (1.1) and \(u_o\).
Our main result concerning the existence of variational solutions which are Lipschitz continuous with respect to the spatial variables can be formulated as follows.
Theorem 1.2
Let \(\Omega \subset \mathbb {R}^n\) be an open, bounded and convex set and \(T \in (0, \infty ]\). Assume that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies hypotheses (1.2). Further, let \(u_o \in W^{1,\infty }(\Omega _T)\) denote a boundary datum such that the bounded slope condition with some positive constant Q (see Definition 2.1 below) is fulfilled for \(U_o:= \left. u_o \right| _{\partial \Omega }\). Then, there exists a unique variational solution u to the Cauchy–Dirichlet problem associated with (1.1) and \(u_o\) in \(\Omega _T\). Moreover, u satisfies the gradient bound
Furthermore, we show that variational solutions to (1.1) are weak solutions and consequently, they are 1/2-Hölder continuous in time provided that the map \(\xi \mapsto f(t, \xi )\) is \(C^1\) and uniformly locally Lipschitz in the following sense: For each \(L>0\), there exists a constant \(M_L>0\) such that
Theorem 1.3
Suppose that the assumptions of Theorem 1.2 hold. Moreover, assume that the mapping \(\xi \mapsto f(t, \xi )\) is in \(C^1({{\mathbb {R}}}^n)\) for almost all \(t \in (0,T)\) and satisfies (1.5). Then the unique variational solution u to the Cauchy–Dirichlet problem associated with (1.1) and \(u_o\) is a weak solution (see (7.1)). Further, it is contained in the space of Hölder continuous functions \(C^{0;1,1/2} ({{{\overline{\Omega }}}}_T)\).
To prove Theorem 1.2, we may assume without a loss of generality that \(T < \infty \), see the beginning of Sect. 6. The proof is divided into three parts. We first assume that the integrand is suitably regular and in particular has a weak derivative with respect to the time variable. Then the method of minimizing movements yields a solution u to the so called gradient constrained obstacle problem to (1.1), where the \(L^\infty \)-norms of the gradients of the solution and the comparison maps are bounded by a fixed constant \(L \in (0, \infty )\). Moreover, the regularity assumption on f ensures that u has a weak time derivative in \(L^2(\Omega _T)\).
Next, under the same regularity assumptions on f as in the first step, a standard argument exploiting the bounded slope condition and the maximum principle yields the uniform gradient bound (1.4) for u. Choosing L large enough, this in turn allows us to deduce that u is in fact already a solution to the unconstrained problem in the sense of Definition 1.1.
To deal with a general integrand f, we consider its Steklov average \(f_\varepsilon \). Since \(f_\varepsilon \) admits a weak time derivative, by the results mentioned in the preceding paragraph there exists a solution \(u_\varepsilon \) to the Cauchy–Dirichlet problem associated with \(f_\varepsilon \) in the sense of Definition 1.1. Moreover, since for each \(\varepsilon >0\) the solution \(u_\varepsilon \) satisfies the gradient bound (1.4) and \(u_\varepsilon \) = \(u_o\) on \(\partial \Omega \times (0,T)\), there exists a limit map \(u \in L^\infty (\Omega _T)\) such that \(u_\varepsilon \rightarrow u\) uniformly and \(Du_\varepsilon \overset{\raisebox {-1ex}{*}}{\rightharpoondown }Du\) weakly\(^*\) up to a subsequence as \(\varepsilon \downarrow 0\). This allows us to conclude that u is a variational solution in the sense of Definition 1.1, finishing the proof of Theorem 1.2.
The proof of Theorem 1.3 is similar to the one found in [7, Chapter 8]. The \(C^1\) assumption on the integrand ensures the validity of the weak Euler–Lagrange equation, which lets us apply the argument from [6, pp. 23–24] to prove a Poincaré inequality for variational solutions. The Hölder continuity then follows from the Campanato space characterization of Hölder continuity by Da Prato [9].
The paper is organized as follows. Section 2 contains preliminary definitions and basic observations about the integrand. In Sect. 3 we prove certain properties of variational solutions that are required in later sections, including the comparison and maximum principles. Under additional regularity assumptions on f we use the method of minimizing movements to prove the existence of variational solutions to the gradient constrained problem in Sect. 4 and in Sect. 5 we consider the unconstrained problem. Finally, in Sect. 6 we consider general integrands and finish the proof of Theorem 1.2 and Hölder continuity in time is proven in Sect. 7 under additional regularity assumptions.
2 Preliminaries
2.1 Notation
Throughout the paper, for \(p \in [1,\infty ]\) and \(m \in \mathbb {N}\) the space \(L^p(\Omega ,\mathbb {R}^m)\) denotes the usual Lebesgue space (we omit \(\mathbb {R}^m\) if \(m=1\)) and \(W^{1,p}(\Omega )\) and \(W^{1,p}_0(\Omega )\) denote the usual Sobolev spaces. If \(\Omega \) is a bounded Lipschitz domain, \(W^{1,\infty }(\Omega )\) can be identified with the space \(C^{0,1}({\overline{\Omega }})\) of functions \(v :\Omega \rightarrow \mathbb {R}\) that are Lipschitz continuous (with Lipschitz constant \([v]_{0,1} = \Vert Dv\Vert _{L^\infty (\Omega ,\mathbb {R}^n)}\)) up to the boundary of \(\Omega \). Note that in particular any convex set has a Lipschitz continuous boundary, since convex functions are locally Lipschitz [11, Corollary 2.4]. Further, for a Banach space X and an integrability exponent \(p \in [1,\infty ]\) we write \(L^p(0,T;X)\) for the space of Bochner measurable functions \(v :[0,T] \rightarrow X\) with \(t \mapsto \Vert v(t)\Vert _X \in L^p(0,T)\). Moreover, \(C^0([0,T];X)\) is defined as the space of the continuous functions \(v :[0,T] \rightarrow X\). For maps v defined in \(\Omega _T\) we also use the short notation v(t) for the partial map \(x \mapsto v(x,t)\) defined in \(\Omega \). Finally, for a set \(A \subset \mathbb {R}^m\), the characteristic function \(\chi _A :\mathbb {R}^m \rightarrow \{0,1\}\) is given by \(\chi _A(x) = 1\) if \(x \in A\) and \(\chi _A(x) = 0\) else.
2.2 Bounded slope condition
In the proof of the existence result in Sect. 5 it is crucial that there exist affine comparison functions below and above the initial/boundary datum \(u_o\) coinciding with \(u_o\) in a point \(x_o \in \partial \Omega \). This is ensured by applying the following bounded slope condition to \(\left. u_o \right| _{\partial \Omega }\).
Definition 2.1
A function \(U :\partial \Omega \rightarrow \mathbb {R}\) satisfies the bounded slope condition with constant \(Q>0\) if for any \(x_o \in \partial \Omega \) there exist two affine functions \(w_{x_o}^\pm :\mathbb {R}^n \rightarrow \mathbb {R}\) with Lipschitz constants \([w_{x_o}^\pm ]_{0,1} \le Q\) such that
Note that unless U itself is affine, the convexity of \(\Omega \) is necessary for the bounded slope condition to hold. Even strict convexity of \(\Omega \) is not sufficient for general U, since the boundary can become “too flat”. However, we know that for a uniformly convex, bounded \(C^2\)-domain \(\Omega \) and \(v \in C^2(\mathbb {R}^n)\) the restriction \(U= \left. v \right| _{\partial \Omega }\) fulfills the bounded slope condition. For more details, we refer to [14, 26]. On the other hand, in the parabolic setting the following example is relevant: Consider a convex domain \(\Omega \) with flat parts (such as a rectangle) and a Lipschitz continuous function \(u_o\) that vanishes at the boundary of \(\Omega \); i.e. we prescribe zero lateral boundary values, but the initial datum is not necessarily identical to zero.
We need the following lemma from [7, Lemma 2.3]. It states that if \(u_o\) is Lipschitz and \(u_o|_{\partial \Omega }\) satisfies the bounded slope condition, then \(u_o\) can be squeezed between two affine functions that touch \(u_o\) at a given boundary boundary point and the Lipschitz constant of these affine functions is bounded by either the Lipschitz constant of \(u_o\) or the constant in the bounded slope condition.
Lemma 2.2
Let \(u_o \in C^{0,1}({\overline{\Omega }})\) with Lipschitz constant \([u_o]_{0,1} \le Q_1\) such that the restriction \(U:= \left. u_o \right| _{\partial \Omega }\) satisfies the bounded slope condition with constant \(Q_2\). Then for any \(x_o \in \partial \Omega \) there exist two affine functions \(w_{x_o}^\pm \) with \([w_{x_o}^\pm ]_{0,1} \le \max \{Q_1, Q_2\}\) such that
2.3 Dominating functions for the integrand
Observe that for any \(L>0\) the map \(t \mapsto \max _{|\xi | \le L} f(t,\xi )\) is measurable, since we have that \(\max _{|\xi | \le L} f(t,\xi ) = \max _{\xi \in B_L(0) \cap \mathbb {Q}^n} f(t, \xi )\) and the maximum of countably many measurable functions is measurable. The same holds true for \(t \mapsto \min _{|\xi | \le L} f(t,\xi )\). In the following lemma, we show that they are contained in \(L^1(0,T)\).
Lemma 2.3
Let \(T\in (0, \infty )\) and assume that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies (1.2). Then, for any \(L>0\) there exists a function \(g_L \in L^1(0,T)\) such that
Proof
First, we show that for any \(L>0\), we have that
To this end, fix \(\xi _1, \ldots , \xi _{n+1} \in \mathbb {R}^n\) such that the closed ball \(B_L(0)\) is a subset of the simplex
Note that for any \(t \in [0,T]\) such that \(\mathbb {R}^n \ni \xi \mapsto f(t,\xi )\) is convex, the mapping \(\xi \mapsto f(t,\xi )\) attains its maximum in one of the points \(\xi _1, \ldots , \xi _{n+1}\). Hence, for a.e. t we obtain that
Since the maps \(t \mapsto f(t,0)\) and \(t \mapsto f(t,\xi _i)\), \(i=1,\ldots ,n+1\), belong to \(L^1(0,T)\) by (1.2)\(_2\), this implies (2.2).
Next, we fix \(L>0\) and prove
Consider \(t \in [0,T]\) such that \(\xi \mapsto f(t,\xi )\) is convex. Then, there exist \(\xi _{min}, \xi _{max} \in B_L(0)\) such that \(f(t,\xi _{min}) = \min _{|\xi | \le L} f(t,\xi )\) and \(f(t,\xi _{max}) = \max _{|\xi | \le L} f(t,\xi )\). Assume that \(\xi _{min} \ne \xi _{max}\) (otherwise, \(\xi \mapsto f(t,\xi )\) is constant in \(B_L(0)\) and thus \(f(t,0) = \min _{|\xi |\le L)} f(t, \xi )\)). First, note that for \(C:= \tfrac{1}{2\,L} (f(t,\xi _{max}) - f(t,\xi _{min})) \in (0,\infty )\), we find that
Furthermore, since \(\xi \mapsto f(t,\xi )\) is convex in \(\mathbb {R}^n\), its subdifferential at \(\xi _{max}\) is non-empty [11, Proposition 5.2], i.e. there exists \(\eta = \eta (\xi _{max}) \in \mathbb {R}^n\) such that
for any \(\xi \in \mathbb {R}^n\). In particular, we have that
where \(\alpha \) denotes the angle between \(\eta \) and \(\xi _{min} - \xi _{max}\). Together, the preceding two inequalities imply that
Next, choose \(s>1\) such that \(\xi _o:= \xi _{min} + s(\xi _{max} - \xi _{min}) \in \partial B_{L+1}(0)\). Note that the vector \(\xi _o - \xi _{max} = (1-s)(\xi _{min} - \xi _{max})\) points in the opposite direction as \(\xi _{min} - \xi _{max}\). Therefore, the angle between \(\eta \) and \(\xi _o - \xi _{max}\) is \(\pi - \alpha \). Using the facts that \(\cos (\pi - \alpha ) = -\cos (\alpha )\) and \(|\xi _o - \xi _{max}| \ge 1\), the preceding inequality and the definition of C, we conclude that
This is equivalent to
which holds for almost every \(t \in [0, T]\). Since we have already shown that \(t \mapsto \max _{|\xi | \le L} f(t,\xi )\) and \(t \mapsto \max _{|\xi | \le L+1} f(t,\xi )\) are contained in \(L^1(0,T)\), the preceding inequality proves (2.3). The claim of Lemma 2.3 follows by combining (2.2) and (2.3). \(\square \)
2.4 Lower semicontinuity
In the course of the paper we will need the following result on the lower semicontinuity of integrals involving f with respect to the weak\(^*\) topology of \(L^\infty (\Omega _T,\mathbb {R}^n)\).
Lemma 2.4
Let \(\Omega \subset \mathbb {R}^n\) be a bounded open set and \(0<T<\infty \). Assume that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies (1.2). Then, for any sequence \((V_i)_{i \in \mathbb {N}} \subset L^\infty (\Omega _T,\mathbb {R}^n)\) and \(V \in L^\infty (\Omega _T,\mathbb {R}^n)\) such that \(V_i \overset{\raisebox {-1ex}{*}}{\rightharpoondown }V\) weakly\(^*\) in \(L^\infty (\Omega _T,\mathbb {R}^n)\) as \(i \rightarrow \infty \) we have that
Proof
Consider an arbitrary sequence \((V_i)_{i \in \mathbb {N}} \subset L^\infty (\Omega _T,\mathbb {R}^n)\) and a limit map \(V \in L^\infty (\Omega _T,\mathbb {R}^n)\) such that \(V_i \overset{\raisebox {-1ex}{*}}{\rightharpoondown }V\) weakly\(^*\) in \(L^\infty (\Omega _T,\mathbb {R}^n)\) as \(i \rightarrow \infty \). First, note that \((V_i)_{i \in \mathbb {N}}\) is bounded in \(L^\infty (\Omega _T,\mathbb {R}^n)\) and set \(M := \sup _{i \in \mathbb {N}} \Vert V_i \Vert _{L^\infty (\Omega _T,\mathbb {R}^n)} \ge \Vert V \Vert _{L^\infty (\Omega _T,\mathbb {R}^n)}\). We find that
is a convex subset of \(L^2(\Omega _T,\mathbb {R}^n)\). Therefore, since \(\xi \mapsto f(t,\xi )\) is convex for a.e. \(t \in [0,T]\) and since \(\iint _{\Omega _T} f(t,W) \,\textrm{d}x\textrm{d}t\) is finite for any \(W \in C\) by (2.1), we obtain that the functional \(F :L^2(\Omega _T,\mathbb {R}^n) \rightarrow (-\infty ,\infty ]\) given by
is proper and convex. Further, F is lower semicontinuous with respect to the norm topology in \(L^2(\Omega _T,\mathbb {R}^n)\). Indeed, assume that the sequence \((W_i)_{i \in \mathbb {N}} \subset L^2(\Omega _T,\mathbb {R}^n)\) converges strongly in \(L^2(\Omega _T,\mathbb {R}^n)\) to a limit map \(W \in L^2(\Omega _T,\mathbb {R}^n)\) as \(i \rightarrow \infty \). If \(\liminf _{i \rightarrow \infty } F[W_i] = \infty \), the assertion \(F[W] \le \liminf _{i \rightarrow \infty } F[W_i]\) holds trivially. Otherwise, there exists a subsequence \({\mathfrak {K}} \subset \mathbb {N}\) such that \(W_i \in C\) for any \(i \in {\mathfrak {K}}\), \(\liminf _{i \rightarrow \infty } F[W_i] = \lim _{{\mathfrak {K}} \ni i \rightarrow \infty } F[W_i]\) and \(W_i \rightarrow W\) a.e. in \(\Omega _T\) as \({\mathfrak {K}} \ni i \rightarrow \infty \). By (2.1) and the dominated convergence theorem, we conclude that \(F[W] = \lim _{{\mathfrak {K}} \ni i \rightarrow \infty } F[W_i] = \liminf _{i \rightarrow \infty } F[W_i]\). Therefore, F is also lower semicontinuous with respect to the weak topology in \(L^2(\Omega _T,\mathbb {R}^n)\), cf. [11, Corollary 2.2]. Since \(\Omega _T\) is bounded, we have that \(V_i \rightharpoondown V\) weakly in \(L^2(\Omega _T,\mathbb {R}^n)\) as \(i \rightarrow \infty \) and hence
This concludes the proof of the lemma. \(\square \)
2.5 Steklov averages of the integrand
For the final approximation argument in the proof of Theorem 1.2 we need to regularize the integrand f with respect to time. To this end, extend f to \([0,\infty ] \times \mathbb {R}^n\) by zero if \(T<\infty \). For \(\varepsilon >0\) define the Steklov average \(f_\varepsilon :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) of the extended integrand by
In order to investigate convergence of the Steklov averages as \(\varepsilon \downarrow 0\), first note that specializing the proof of [11, Corollary 2.4] gives us the following result.
Lemma 2.5
Let \(L>0\) and assume that \(f :\mathbb {R}^n \rightarrow \mathbb {R}\) is a convex function with \(\Vert f\Vert _{L^\infty (B_{L+1}(0))} \le C\). Then, f satisfies the local Lipschitz continuity condition
We also need the following variant of the dominated convergence theorem that can be found for example in [12, Theorem 1.20].
Lemma 2.6
Assume that \(v, v_k \in L^1(\mathbb {R}^n)\) and \(w, w_k \in L^1(\mathbb {R}^n)\) are measurable for all \(k \in \mathbb {N}\). Suppose that \(w_k \rightarrow w\) a.e. in \(\mathbb {R}^n\) and \(|w_k| \le v_k\) for all \(k \in \mathbb {N}\). Suppose moreover that \(v_k \rightarrow v\) a.e. in \(\mathbb {R}^n\) and
Then
With the preceding lemmas at hand, we prove the following convergence result.
Lemma 2.7
Let \(T \in (0,\infty )\) and assume that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies hypotheses (1.2). For \(\varepsilon >0\) let \(f_\varepsilon :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) denote the Steklov average of f given by (2.4). Then, we have that
Proof
Fix \(L>0\). First of all, we show that
By (1.2)\(_2\), for fixed \(\xi \in \mathbb {R}^n\) we have that \(f_\varepsilon (t,\xi ) \rightarrow f(t,\xi )\) for a.e. \(t \in [0,T]\) by Lebesgue’s differentiation theorem. Thus, there exists a set N of \(\mathcal {L}^1\)-measure zero such that
Without loss of generality assume that additionally for all \(t \in [0,T] {\setminus } N\) the map \(\xi \mapsto f(t,\xi )\) is convex, the function \(g_{L+1}\) from (2.1) fulfills \(g_{L+1}(t) < \infty \) and there holds 
. Now, fix \(t \in [0,T] {\setminus } N\). By (2.1) and Lemma 2.5 we conclude that \(\xi \mapsto f(t,\xi )\) is Lipschitz continuous in \(B_L(0)\) with Lipschitz constant \(2 g_{L+1}(t)\). Using this together with the definition of the Steklov average, for any \(\xi _1, \xi _2 \in B_L(0)\) we compute that
Since 
, there exists \(\varepsilon _o>0\) such that \(\xi \mapsto f_\varepsilon (t,\xi )\) is Lipschitz continuous with Lipschitz constant \(4 g_{L+1}(t)\) for all \(\varepsilon \in (0,\varepsilon _o]\). This shows that the sequence \((f_\varepsilon (t,\cdot ))_{\varepsilon \in (0,\varepsilon _o]}\) is equicontinuous in \(B_L(0)\). Moreover, \((f_\varepsilon (t,\cdot ))_{\varepsilon \in (0,\varepsilon _o]}\) is equibounded in \(B_L(0)\), since for any \(\xi \in B_L(0)\) and \(\varepsilon \in (0,\varepsilon _o]\), we find that
Therefore, we infer from the Arzèla–Ascoli theorem that \((f_\varepsilon (t,\cdot ))_{\varepsilon \in (0,\varepsilon _o]}\) converges uniformly in \(B_L(0)\) as \(\varepsilon \downarrow 0\) and the limit \(f(t,\cdot )\) is determined by (2.6). This concludes the proof of (2.5). Next, since
where 
in \(L^1(0,T)\), the claim now follows from Lemma 2.6. \(\square \)
2.6 Mollification in time
In general, variational solutions are not admissible as comparison maps in the variational inequality (1.3), since they do not necessarily admit a derivative with respect to time. Therefore, we use the following mollification procedure with respect to time. More precisely, consider a separable Banach space X, an initial datum \(v_o \in X\) and a map \(v \in L^r(0,T;X)\) for some \(r \in [1,\infty ]\). For \(h>0\) define the mollification
Later on, we will mainly use \(X= L^q(\Omega )\) or \(X = W^{1,q}(\Omega )\) for some \(q \in [1,\infty )\). A vital feature of this mollification procedure is that \([v]_h\) solves the ordinary differential equation
with initial condition \([v]_h(0) = v_o\). This shows in particular that if v and \([v]_h\) are contained in a function space, the same holds true for the time derivative of \([v]_h\). The basic properties of time mollifications are collected in the following lemma (cf. [19, Lemma 2.2] and [5, Appendix B] for the proofs).
Lemma 2.8
Let X be a separable Banach space and \(v_o \in X\). If \(v \in L^r(0,T;X)\) for some \(r \in [1,\infty ]\), then also \([v]_h \in L^r(0,T;X)\) and if \(r < \infty \), then \([v]_h \rightarrow v\) in \(L^r(0,T;X)\) as \(h \downarrow 0\). Further, for any \(t_o \in (0,T]\) there holds the bound
where the bracket \([\ldots ]^\frac{1}{r}\) has to be interpreted as 1 if \(r = \infty \). Moreover, if \(v \in C^0([0,T];X)\), then also \([v]_h \in C^0([0,T];X)\) with \([v]_h(0) = v_o\) and there holds \([v]_h \rightarrow v\) in \(L^\infty (0,T;X)\) as \(h \downarrow 0\).
For maps \(v \in L^r(0,T;X)\) with \(\partial _t v \in L^r(0,T;X)\) we have the following assertion.
Lemma 2.9
Let X be a separable Banach space and \(r \ge 1\). Assume that \(v \in L^r(0,T;X)\) with \(\partial _t v \in L^r(0,T;X)\). Then, for the mollification in time defined by
the time derivative can be computed by
3 Properties of variational solutions
As mentioned in the introduction, besides variational solutions in the sense of Definition 1.1, we consider variational solutions of the so-called gradient constrained obstacle problem to (1.1). They enjoy the same basic properties as variational solutions to the unconstrained Cauchy–Dirichlet problem to (1.1) and proofs will be given in a unified way in this section.
Let \(L \in (0, \infty ]\). We define the following class of functions that are L-Lipschitz in space
and given time-independent boundary values \(u_o \in W^{1,\infty }(\Omega )\) with \(\Vert Du_o\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L\), we denote the subclass
Definition 3.1
Assume that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies (1.2), consider a boundary datum \(u_o \in W^{1,\infty }(\Omega )\) and let \(L \in (0,\infty )\) be such that \(\Vert Du_o\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L\). In the case \(T<\infty \) a map \(u \in K^L_{u_o}(\Omega _T)\) is called a variational solution to the gradient constrained Cauchy–Dirichlet problem associated with (1.1) and \(u_o\) in \(\Omega _T\) if and only if the variational inequality
holds true for any comparison map \(v \in K^L_{u_o}(\Omega _T)\) with \(\partial _t v \in L^2(\Omega _T)\). If \(T=\infty \) and \(u \in K^L_{u_o}(\Omega _\infty )\) is a variational solution in \(\Omega _\tau \) for any \(\tau > 0\), u is called a global variational solution or variational solution in \(\Omega _\infty \) to the gradient constrained Cauchy–Dirichlet problem associated with (1.1) and \(u_o\).
3.1 Continuity with respect to time
In Definitions 1.1 and 3.1 we require that variational solutions are contained in the space \(C^0([0,T];L^2(\Omega ))\). However, this is already implied if u satisfies a variational inequality for a.e. \(\tau \in [0,T]\). More precisely, we have the following Lemma, which will be applied with \(L=\infty \) in Sect. 6.
Lemma 3.2
Let \(\Omega \subset \mathbb {R}^n\) be open and bounded and \(T\in (0,\infty )\) and assume that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies (1.2). Let \(L \in (0,\infty ]\) and consider \(u_o\in W^{1,\infty }(\Omega )\) such that \(\Vert Du_o\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L\). Further, consider \(u\in L^{\infty }(\Omega _{T})\) with \(u=u_o\) on \(\partial _{\mathcal {P}}\Omega _{T}\) and \(\left\| Du\right\| _{L^{\infty }(\Omega _T,{\mathbb {R}}^{n})} \le L\) if \(L \in (0,\infty )\) and \(\left\| Du\right\| _{L^{\infty }(\Omega _T,{\mathbb {R}}^{n})}<\infty \) if \(L=\infty \), respectively. Suppose that u satisfies the variational inequality
for almost all \(\tau \in (0,T)\) whenever \(v\in K_{u_o}^L(\Omega _{T})\) with \(\partial _{t}v\in L^{2}(\Omega _{T})\). Then, we have that \(u\in C^{0}([0,T];L^{2}(\Omega ))\).
Proof
The proof is similar to that of Lemma 2.6 in [28] except for the estimate of the second integral in (3.3) below. Denote by \([u]_h\) the time mollification of u with initial values \(u_o\) as defined in (2.7). In particular, observe that \([u]_{h}\in C^{0}([0,T];L^{2}(\Omega ))\), since we know that \(\partial _{t}[u]_{h}\in L^{2}(\Omega _{T})\) and \([u]_{h}(0) = u_o \in L^{2}(\Omega )\). Using \([u]_h\) as a comparison function in (3.2), taking the essential supremum over \(\tau \in (0,T)\) and recalling that \(([u]_h - u) = - h \partial _t [u]_h\), we obtain that
Furthermore, we have that \(D[u]_{h}\rightarrow Du\) almost everywhere in \(\Omega _{T}\) as \(h\downarrow 0\) (up to a subsequence) and that \(|D[u]_{h}| \le |Du_o| + \sup _{\Omega _T} |Du|\). Therefore, by (2.1) and the dominated convergence theorem we find that the second integral in (3.3) vanishes in the limit \(h\downarrow 0\). Hence, we have shown that \([u]_{h}\rightarrow u\) in \(L^{\infty }(0,T;L^2(\Omega ))\). Combining this with the fact that \([u]_h \in C^0([0,T];L^2(\Omega ))\), it follows that also \(u\in C^{0}([0,T];L^2(\Omega ))\). \(\square \)
3.2 Localization in time
Here, we show that a variational solution in a space-time cylinder \(\Omega _T\) is also a solution in any sub-cylinder \(\Omega _\tau \), \(\tau \in (0,T)\).
Lemma 3.3
(Localization in time) Let \(T \in (0,\infty )\), assume that \(\Omega \subset {\mathbb {R}}^n\) is open and bounded, and that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies (1.2). Consider \(u_o \in W^{1,\infty }(\Omega )\) and \(L \in (0, \infty ]\) such that \(\Vert Du_o\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L\). Suppose that u is a variational solution to (1.1) in \(K^{L}_{u_o}(\Omega _{T})\) (in the sense of Definition 3.1 if \(L < \infty \), in the sense of Definition 1.1 if \(L = \infty \)). Then \(\left. u\right| _{\Omega _{\tau }}\) is a variational solution to (1.1) in \(K^{L}_{u_o}(\Omega _{\tau })\) for any \(\tau \in (0,T]\).
Proof
For \(\theta \in (0,\tau )\), consider the cut-off function
For \(v\in K_{u_o}^{L}(\Omega _{\tau })\) satisfying \(\partial _{t}v\in L^{2}(\Omega _{\tau })\) we define a function \(v_{\theta } :\Omega _{T}\rightarrow {\mathbb {R}}\) by
where \(\xi _{\theta }v\) has been extended to \(\Omega _{T}\) by zero and \([u]_h\) is defined according to (2.7) with initial datum \(u_o\). Then we have \(v_{\theta }\in K_{u_o}^{L}(\Omega _{T})\) with \(\partial _{t}v_{\theta }\in L^{2}(\Omega _{T})\), and therefore we may use \(v_{\theta }\) as a comparison map for u in the variational inequality. This yields
The first term on the right-hand side of (3.4) is identical to the one in [7, Equation (3.2)] and can be estimated in the same way to obtain
The second term on the right-hand side of (3.4) is given by
Since we know that
by (2.1) we find that
in the limit \(\theta \downarrow 0\). Combining the preceding estimates we arrive at
Note that \([u]_h \rightarrow u\) in \(L^\infty (0,T;L^2(\Omega ))\) as \(h \downarrow 0\), since \(u \in C^0([0,T];L^2(\Omega ))\). Further, we have that \(D[u]_{h}\rightarrow Du\) pointwise almost everywhere in \(\Omega _{T}\) as \(h \downarrow 0\) (up to a subsequence) and that
Therefore, assumption (2.1), the fact that \(\Omega \) is bounded and the dominated convergence theorem imply that
Hence, using that \(\partial _{t}[u]_{h}([u]_{h}-u)\le 0\) and letting \(h\downarrow 0\) in (3.5), we obtain the desired inequality
\(\square \)
3.3 The initial condition
As a consequence of the localization in time principle, we find that variational solutions attain the initial datum \(u_o\) in the \(C^0\)-\(L^2\)-sense. The precise statement is as follows.
Lemma 3.4
Let \(T \in (0,\infty )\), assume that \(\Omega \subset {\mathbb {R}}^{n}\) is bounded and open, and that \(f :[0,T]\times {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\) satisfies (1.2). Consider \(u_o\in W^{1,\infty }(\Omega )\) and \(L\in (0,\infty ]\) such that \(\left\| Du_o\right\| _{L^{\infty }(\Omega ,{\mathbb {R}}^{n})}\le L\). Suppose that u is a variational solution to (1.1) in \(K_{u_o}^{L}(\Omega _{T})\) (in the sense of Definition 3.1 if \(L<\infty \), in the sense of Definition 1.1 if \(L=\infty \)). Then, there holds
Proof
By Lemma 3.3, the function u is a variational solution in any smaller cylinder \(\Omega _{\tau }\), \(\tau \in (0,T]\). Using \(v :\Omega _{\tau }\rightarrow {\mathbb {R}}\), \(v(x,t):=u_o(x)\) as a comparison function for u and taking (2.1) with \(M:= \max \{ \Vert Du\Vert _{L^{\infty }(\Omega _{T},{\mathbb {R}}^{n})}, \Vert Du_o\Vert _{L^{\infty }(\Omega ,{\mathbb {R}}^{n})} \}\) into account, we obtain that
Since \(g_M \in L^1(0,T)\), this implies the claim. \(\square \)
3.4 Comparison principle
The following comparison principle ensures in particular that variational solutions to the problems considered in the present paper are unique.
Theorem 3.5
(Comparison principle) Let \(T \in (0,\infty )\), assume that \(\Omega \subset {\mathbb {R}}^n\) is bounded and open, and that \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies (1.2). Let \(L \in (0, \infty ]\) and suppose that u and \({\tilde{u}}\) are variational solutions to (1.1) in \(K^{L}(\Omega _{T})\) (in the sense of Definition 3.1 if \(L < \infty \) and in the sense of Definition 1.1 if \(L = \infty \)) such that \(\Vert Du(0)\Vert _{L^{\infty }(\Omega ,\mathbb {R}^n)}\) and \(\Vert D{\tilde{u}}(0)\Vert _{L^{\infty }(\Omega ,\mathbb {R}^n)}\) are bounded by L if \(L \in (0,\infty )\) and finite if \(L=\infty \), respectively. Then the assumption that
implies
Proof
Let \(\tau \in (0,T]\). By Lemma 3.3, the functions u and \({\tilde{u}}\) are variational solutions in \(K^{L}(\Omega _{\tau })\). Consider the functions
where \([u]_h\) and \([{\tilde{u}}]_h\) denote the mollifications of u and \({\tilde{u}}\) according to (2.7) with initial values \(u(0) \in W^{1,\infty }(\Omega )\) and \({\tilde{u}}(0) \in W^{1,\infty }(\Omega )\), respectively. Since the boundary values attained by u and \({\tilde{u}}\) are independent of time, we have that \(v\in K_{u}^{L}(\Omega _{\tau })\) and \(w\in K_{{\tilde{u}}}^{L}(\Omega _{\tau })\) with \(\partial _{t}v,\partial _{t}w\in L^{2}(\Omega _{\tau })\). Therefore we may use v and w as comparison functions in the variational inequalities of u and \({\tilde{u}}\), respectively. Adding the resulting inequalities and using that \([u]_h(0) = u(0) \le {\tilde{u}}(0) = [{\tilde{u}}]_h(0)\), we obtain
Using the identities
we compute that
Therefore, taking into account that \([u]_h(0) = u(0) \le {\tilde{u}}(0) = [{\tilde{u}}]_h(0)\), we find that
Furthermore, using \([u]_{h}\) as a comparison function for u and omitting the boundary term at time \(\tau \) on the right-hand side of the variational inequality, we obtain
and a similar inequality holds for \({\tilde{u}}.\) Observe also that
Combining the estimates (3.7), (3.8) and (3.9) with (3.6) we arrive at
By the same argument as in the end of the proof of Lemma 3.3 involving the dominated convergence theorem, the integral on the right-hand side of (3.10) vanishes in the limit \(h \downarrow 0\). Writing \(v-u = -([u]_{h}-[{\tilde{u}}]_{h})_{+} + [u]_h - u\) and \(w - {\tilde{u}} = ([u]_{h}-[{\tilde{u}}]_{h})_{+} + [{\tilde{u}}]_h - {\tilde{u}}\) and using that \([u]_{h}\rightarrow u\) and \([{\tilde{u}}]_h \rightarrow {\tilde{u}}\) in \(L^{\infty }([0,\tau ],L^{2}(\Omega ))\) as \(h \downarrow 0\) since \(u, {\tilde{u}} \in C^0([0,T];L^2(\Omega ))\), we conclude that
Hence, taking the limit \(h \downarrow 0\) in (3.10), we infer
which implies that \(u\le {\tilde{u}}\) in \(\Omega _{\tau }\). Since \(\tau \) was arbitrary, the claim follows. \(\square \)
3.5 Maximum principle and localization in space for regular solutions
In this section, we consider more regular variational solutions u satisfying \(\partial _{t}u \in L^{2}(\Omega _{T})\). As a consequence, u is directly admissible as comparison map in its variational inequality without regularization with respect to the time variable. Further, due to the requirements of the proof of the existence result in Sect. 5, we will take time-dependent boundary values \(\left. u \right| _{\Omega \times (0,T)}\) into account here. In particular, the proof of the comparison principle in Theorem 3.5 is easily adapted to allow time-dependent boundary values if \(\partial _{t}u\) and \(\partial _t {\tilde{u}}\) are contained in \(L^{2}(\Omega _{T})\) by using \(\min (u,{\tilde{u}})\) and \(\max (u,{\tilde{u}})\) as comparison maps in the variational inequalities satisfied by u and \({\tilde{u}}\), respectively, and proceeding in a similar way as above. However, most arguments can be simplified, since mollification with respect to time is not necessary in the present situation. This allows us to deduce the following maximum principle.
Lemma 3.6
(Maximum principle) Let \(T \in (0,\infty )\), assume that \(\Omega \subset {\mathbb {R}}^{n}\) is open and bounded, and that \(f :[0,T]\times {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\) satisfies (1.2). Consider \(L\in (0,\infty ]\) and functions \(u,{\tilde{u}}\in K^{L}(\Omega _{T})\) such that \(\partial _{t}u,\partial _{t}{\tilde{u}}\in L^{2}(\Omega _{T})\). Suppose moreover that \(\left\| Du(0)\right\| _{L^{\infty }(\Omega ,{\mathbb {R}}^{n})}\) and \(\left\| D{\tilde{u}}(0)\right\| _{L^{\infty }(\Omega ,{\mathbb {R}}^{n})}\) are bounded by L if \(L \in (0,\infty )\) and finite if \(L=\infty \). Finally, assume that for any \(\tau \in (0,T]\) the function u satisfies the variational inequality
whenever \(v\in K^{L}(\Omega _{\tau })\) with \(\partial _{t}v\in L^{2}(\Omega _{\tau })\) and \(v=u\) on \(\Omega \times (0,\tau )\), and that \({\tilde{u}}\) fulfills the analogical inequality. Then
Proof
Let \(\tau \in (0,T]\). Define
Then \({\hat{u}}\) satisfies the variational inequality (3.11) with its own boundary values, and
Consider the functions \(v:=\min (u,{\hat{u}})\) and \(w:=\max (u,{\hat{u}})\). Then \(v,w\in K^{L}(\Omega _{\tau })\) with \(\partial _t v, \partial _t w \in L^2(\Omega _\tau )\) and \(v=u\), \(w={\hat{u}}\) on \(\partial \Omega \times (0,\tau )\). Observe also that \(v-u=-(u-{\hat{u}})_{+}\) and \(w-{\hat{u}}=(u-{\hat{u}})_{+}\). Using v and w as comparison functions for u and \({\hat{u}}\) in the variational inequality (3.11), we obtain
where we used that \((v-u)(0)=(w-{\hat{u}})(0)=0\) and that the terms with f cancel one another. As \(\tau \) was arbitrary, we obtain
so that
Since the reverse inequality holds by continuity, this proves the claim. \(\square \)
Lemma 3.7
(Localization in space) Let \(T \in (0,\infty )\), assume that \(\Omega \subset {\mathbb {R}}^{n}\) is open and bounded, and that \(f :[0,T]\times {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\) satisfies (1.2). Consider \(u_o\in W^{1,\infty }(\Omega )\) and \(L\in (0,\infty ]\) such that \(\left\| Du_o\right\| _{L^{\infty }(\Omega ,{\mathbb {R}}^{n})}\le L\). Suppose that u is a variational solution to (1.1) in \(K_{u_o}^{L}(\Omega _{T})\), \(L\in (0,\infty ]\) (in the sense of Definition 3.1 if \(L<\infty \), in the sense of Definition 1.1 if \(L=\infty \)). Moreover, suppose that \(\partial _{t}u\in L^{2}(\Omega _{T})\). Then for any domain \(\Omega ^{\prime }\subset \Omega \) and any \(\tau \in (0,T]\), the variational inequality
holds whenever \(v\in K^{L}_{u_o}(\Omega _{\tau }^{\prime })\) with \(\partial _{t}v\in L^{2}(\Omega _{\tau })\) and \(v=u\) on \(\partial \Omega ^\prime \times (0,\tau )\).
Proof
By Lemma 3.3 the function \(u|_{\Omega _\tau }\) is a variational solution to (1.1) in the function space \(K_{u_o}^L(\Omega _\tau )\). Observe that
is an admissible comparison function for \(u|_{\Omega _\tau }\) in the variational inequality. Inserting w into the variational inequality (3.1) if \(L<\infty \) (or (1.3) if \(L=\infty \)) with T replaced by \(\tau \) immediately yields (3.13). \(\square \)
4 Existence for the gradient constrained problem for regular integrands
In this section, we are concerned with integrands that admit a time derivative. More precisely, we consider \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) such that
The aim of this section is to prove the following existence result.
Theorem 4.1
Let \(\Omega \subset \mathbb {R}^n\) be a bounded Lipschitz domain and \(T \in (0,\infty )\). Consider a boundary datum \(u_o \in W^{1,\infty }(\Omega )\) such that \(\Vert Du_o\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L\) for a constant \(L \in (0, \infty )\). Further, assume that the integrand \(f :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) satisfies hypothesis (4.1). Then, there exists a variational solution \(u \in K^L_{u_o}(\Omega _T)\) to the gradient constrained problem in the sense of Definition 3.1. Further, there holds \(\partial _t u \in L^2(\Omega _T)\) with the quantitative bound
We prove Theorem 4.1 via the method of minimizing movements. The proof is divided into five steps.
4.1 A sequence of minimizers to elliptic variational functionals
Fix a step size \(h:= \frac{T}{m}\) for some \(m \in \mathbb {N}\) and consider times ih, \(i = 0,\ldots ,m\). For \(i=0\), set \(u_0:= u_o \in W^{1,\infty }(\Omega )\) with \(\Vert Du_o\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L\). Further, for \(i = 1,\ldots ,m\), \(u_i\) is defined as the minimizer of the elliptic variational functional
in the class \(\mathcal {A}:= \{v \in W^{1,\infty }(\Omega ): v=u_o \text { on} \partial \Omega \text { and} \Vert Dv\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L \}\). The existence of a minimizer to \(F_i\) in this class is ensured by the direct method in the calculus of variations. More precisely, note that \(\mathcal {A} \ne \emptyset \), since \(u_o \in \mathcal {A}\), and consider a minimizing sequence to \(F_i\) in \(\mathcal {A}\), i.e. a sequence \((u_{i,j})_{j \in \mathbb {N}} \subset \mathcal {A}\) such that
Further, by definition of \(\mathcal {A}\) and Rellich’s theorem there exists a limit map \(u_i \in \mathcal {A}\) and a (not relabelled) subsequence such that
Since the functional \({\widetilde{F}}_i :W^{1,2}(\Omega ) \rightarrow (-\infty ,\infty ]\),
is proper, convex and lower semicontinuous with respect to strong convergence in \(W^{1,2}(\Omega )\), it is also lower semicontinuous with respect to weak convergence in \(W^{1,2}(\Omega )\), see [11, Corollary 2.2]. Therefore, we obtain that
4.2 Energy estimates
Since \(u_{i-1} \in \mathcal {A}\) is an admissible comparison map for the minimizer \(u_i\) and f fulfills (4.1)\(_3\), we have that
Summing up the preceding inequalities from \(i=1\) to \(i=m\), we find that
Subtracting the first term on the left-hand side, we conclude that
4.3 The limit map
In the following we denote the step size by \(h_m\) in order to emphasize the dependence on m. First, we join the minimizers \(u_i\) to a map that is piecewise constant with respect to time. More precisely, we define \(u^{(m)} :\Omega \times (-h_m,T] \rightarrow \mathbb {R}\) by
Observe that the sequence \(\big (u^{(m)} \big )_{m \in \mathbb {N}}\) is bounded in \(L^\infty (\Omega _T)\), since \(\Vert u^{(m)}\Vert _{L^\infty (\Omega _T)} = \max _{i=0,\ldots ,m} \Vert u_i\Vert _{L^\infty (\Omega )}\), \(u_i \in \mathcal {A}\) for all \(i=0,\ldots ,m\) and \(\mathcal {A}\) is equibounded. Further, we know that \(\Vert Du^{(m)}\Vert _{L^\infty (\Omega _T,\mathbb {R}^n)} = \max _{i=0,\ldots ,m} \Vert Du_i\Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \le L\) for any \(m \in \mathbb {N}\). Therefore, there exists a subsequence \({\mathfrak {K}} \subset \mathbb {N}\) and a limit map \(u \in L^\infty (\Omega _T)\) such that \(\Vert Du\Vert _{L^\infty (\Omega _T,\mathbb {R}^n)} \le L\), \(u= u_o\) on \(\partial \Omega \times (0,T)\) and
In order to prove that u has a time derivative, we consider the linear interpolation of minimizers \({\tilde{u}}^{(m)} :\Omega \times (-h_m,T] \rightarrow \mathbb {R}\) given by \({\tilde{u}}^{(m)}(t):= u_o\) for \(t \in (-h_m,0]\) and
Similar arguments as above ensure that \(\big ({\tilde{u}}^{(m)} \big )_{m \in \mathbb {N}}\) is bounded in \(L^\infty (\Omega _T)\) and that \(\Vert D{\tilde{u}}^{(m)}\Vert _{L^\infty (\Omega _T,\mathbb {R}^n)} \le L\) for any \(m \in \mathbb {N}\). Moreover, by the energy bound (4.2) we obtain that
Hence, \(\big ( {\tilde{u}}^{(m)} \big )_{m \in \mathbb {N}}\) is bounded in \(W^{1,2}(\Omega _T)\). By Rellich’s theorem we conclude that there exists a subsequence still labelled \({\mathfrak {K}}\) and a limit map \({\tilde{u}} \in L^\infty (\Omega _T)\) with \(\Vert D{\tilde{u}}\Vert _{L^\infty (\Omega _T,\mathbb {R}^n)} \le L\), \({\tilde{u}}= u_o\) on \(\partial \Omega \times (0,T)\) and \(\partial _t {\tilde{u}} \in L^2(\Omega _T)\) such that
Note that \(\partial _t {\tilde{u}} \in L^2(\Omega _T)\) in particular implies that \({\tilde{u}} \in C^{0;\frac{1}{2}}([0,T];L^2(\Omega ))\) and therefore \({\tilde{u}}\) is contained in the class of functions \(K_{u_o}^L(\Omega _T)\). Next, since \(\big | \big ( u^{(m)} - {\tilde{u}}^{(m)} \big )(t) \big | \le |u_i - u_{i-1}|\) for \(t \in ((i-1)h_m, ih_m]\), \(i=1,\ldots ,m\), we infer from (4.2) that
Together with (4.5)\(_1\) this implies that \(u^{(m)} \rightarrow {\tilde{u}}\) strongly in \(L^2(\Omega _T)\) as \({\mathfrak {K}} \ni m \rightarrow \infty \) and thus in particular that \(u = {\tilde{u}} \in K_{u_o}^L(\Omega _T)\) with \(\partial _t u \in L^2(\Omega _T)\). Finally, by lower semicontinuity with respect to weak convergence, (4.4) gives us the claimed bound
4.4 Minimizing property of the approximations
First, define piecewise constant approximations of the integrand by
We claim that \(u^{(m)}\) is a minimizer of the functional
in the class of functions
Indeed, consider an arbitrary map \(v \in \mathcal {A}_T\). Since \(v(t) \in \mathcal {A}\) for a.e. \(t \in [0,T]\), by the minimizing property of \(u_i\) with respect to \(F_i\) in the class \(\mathcal {A}\) we find that
A straightforward computation shows that this is equivalent to
for any \(v \in \mathcal {A}_T\). Choosing the convex combination \(u^{(m)} + s \big ( v - u^{(m)} \big ) \in \mathcal {A}_T\) with \(s \in (0,1)\) as comparison map and using the convexity of \(\xi \mapsto f(t,\xi )\) for all \(t \in [0,T]\), we obtain that
Reabsorbing the first term on the right-hand side into the left-hand side, dividing the resulting inequality by s and taking the limit \(s \downarrow 0\) gives us that
Next, assume without loss of generality that \(v(0) \in L^\infty (\Omega )\), extend v to \((-h_m, 0]\) by v(0) and note that
Inserting this into the preceding inequality and recalling that \(v(t) = v(0)\) for \(t \in (-h_m, 0]\), we infer
4.5 Variational inequality for the limit map
We fix an arbitrary map \(v \in K^L_{u_o}(\Omega _T)\) with \(\partial _t v \in L^2(\Omega _T)\). Thus, in particular we have that \(v \in \mathcal {A}_T\), so v is an admissible comparison map in (4.6). Our goal is to pass to the limit \({\mathfrak {K}} \ni m \rightarrow \infty \) in (4.6) in order to deduce the variational inequality (3.1) for u. To this end, we consider the terms separately. First, we write the first term on the left-hand side of (4.6) as
By Lemma 2.4 and (4.3)\(_3\), we obtain that
Further, since \(\big \Vert Du^{(m)} \big \Vert _{L^\infty (\Omega _T,\mathbb {R}^n)} \le L\) for all \(m \in \mathbb {N}\) and f fulfills (4.1)\(_3\), we estimate
Therefore, this term vanishes in the limit \(m \rightarrow \infty \). Joining the preceding estimates, we conclude that
Repeating the estimates in the penultimate inequality with \(u^{(m)}\) replaced by v, for the first term on the right-hand side of (4.6) we find that
Next, since \(\frac{1}{h_m} (v(t) - v(t-h_m)) \rightarrow \partial _t v\) strongly in \(L^2(\Omega _T)\) and \(u^{(m)} \rightharpoondown u\) weakly in \(L^2(\Omega _T)\) as \({\mathfrak {K}} \ni m \rightarrow \infty \) by (4.3)\(_1\), we have that
Finally, by the fact that \(v \in C^0([0,T];L^2(\Omega ))\) and by (4.3)\(_2\), we obtain that
Collecting the assertions (4.7)–(4.10) yields
Since \(v \in K^L_{u_o}(\Omega _T)\) with \(\partial _t v \in L^2(\Omega _T)\) was arbitrary, we have shown that \(u \in K^L_{u_o}(\Omega _T)\) is the desired variational solution. \(\square \)
5 Existence for the unconstrained problem for regular integrands
In this section we show the existence of variational solutions to the unconstrained problem under the regularity condition (4.1) provided that the initial and boundary datum satisfies the bounded slope condition. To this end, we need the following lemma, whose proof is similar to that of [7, Lemma 7.1]. It states that affine functions independent of time are variational solutions to (1.1) with respect to their own initial and lateral boundary values.
Lemma 5.1
Let \(\Omega \) be open and bounded. Assume that \(f :[0,T]\times {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) satisfies (1.2). Let \(w(x,t):=a+\xi \cdot x\) with constants \(a\in {\mathbb {R}}\) and \(\xi \in {\mathbb {R}}^{n}\) be an affine function independent of time. Then w is a variational solution in the sense of Definition 1.1 in \(K_{w}^{\infty }(\Omega _{T})\).
With the preceding lemma at hand, we are able to prove the following.
Theorem 5.2
Let \(T \in (0,\infty )\), assume that \(\Omega \subset {\mathbb {R}}^{n}\) is open, bounded and convex, and that the integrand \(f :[0,T]\times {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}\) satisfies (4.1). Consider \(u_o\in W^{1,\infty }(\Omega )\) such that \(\left\| Du_o\right\| _{L^{\infty }(\Omega ,{\mathbb {R}}^{n})}\le Q\) and suppose that \(\left. u_o \right| _{\partial \Omega }\) satisfies the bounded slope condition with the same parameter Q. Then there exists a variational solution \(u\in K^{\infty }_{u_o}(\Omega _{T})\) to (1.1) in the sense of Definition 1.1. Further, we have the quantitative bound
Proof
Let \(L > Q\). By Theorem 4.1 there exists a variational solution \(u \in K^L_{u_o} (\Omega _T)\) with \(\partial _t u \in L^2(\Omega _T)\) to the gradient constrained problem in the sense of Definition 3.1. We begin by proving the Lipschitz bound (5.1) and then show that u is in fact already a solution to the unconstrained problem.
Fix \(x_o \in \partial \Omega \) and denote by \(w_{x_o}^{\pm }\) the affine functions from Lemma 2.2. In particular we have \(w_{x_o}^{-}\le u_o\le w_{x_o}^{+}\). Since by Lemma 5.1 the functions \(w_{x_o}^{-}\) and \(w_{x_o}^{+}\) are variational solutions, it follows from the comparison principle in Theorem 3.5 that
Consequently, there holds
Since \(x_o\in \partial \Omega \) was arbitrary, we obtain that
Consider \(x_{1},x_{2}\in \Omega \), \(x_{1}\not =x_{2}\), \(t\in (0,T)\) and set \(y:=x_{2}-x_{1}\). Define the shifted set \({\widetilde{\Omega }}_{T}:=\left\{ (x-y,t)\in {\mathbb {R}}^{n+1}:(x,t)\in \Omega _{T}\right\} \) and the shifted function \(u_{y}:{\widetilde{\Omega }}_{T}\rightarrow {\mathbb {R}}\) by
Then \(u_{y}\) is a variational solution in \(K^{L}({\widetilde{\Omega }}_{T})\). Since \(\partial _{t}u,\partial _{t}u_{y}\in L^{2}((\Omega \cap {\widetilde{\Omega }})_{T})\) by the spatial localization principle in Lemma 3.7, the functions u and \(u_{y}\) both satisfy variational inequality (3.11) from Lemma 3.6 in \((\Omega \cap {\widetilde{\Omega }})_T\). Therefore by Lemma 3.6 there exists \((x_o,t_o)\in \mathcal {\partial _{\mathcal {P}}}((\Omega \cap {\widetilde{\Omega }}))_{T}\) such that
By definition of y and \(u_{y}\), this yields
Since either \(t_o=0\) or one of the points \(x_o\) or \(x_o+y\) belongs to \(\partial \Omega \), it follows from the assumption \(\left\| Du_o\right\| _{L^{\infty }(\Omega ,{\mathbb {R}}^{n})}\le Q\) and (5.2) that
Combining this with the preceding estimate, we obtain (5.1).
It remains to show that u is a variational solution to the unconstrained problem. Let \(w\in K^\infty _{u_o}(\Omega _{T})\) with \(\partial _{t}w\in L^{2}(\Omega _{T})\) and choose the comparison map \(v:=u+s(w-u)\) for \(0<s\ll 1\); in particular, since \(Q < L\), for s small enough we have that
Thus v is an admissible comparison function for the gradient constrained problem and we obtain that
Reabsorbing the integral with f(t, Du) to the left-hand side and dividing by s, we see that u satisfies the variational inequality (1.3). Thus u is a variational solution in the sense of Definition 1.1. \(\square \)
6 Existence for the unconstrained problem for general integrands
In this section we finish the proof of Theorem 1.2. Note that we only need to consider the case \(T < \infty \). Indeed, assume that for any \(\tau \in (0,\infty )\) we have constructed a variational solution with initial and boundary datum \(u_o\) in the sense of Definition 1.1 such that the gradient bound (1.4) holds in \(\Omega _\tau \). Let \(0< \tau _1< \tau _2 < \infty \) and denote by \(u_1\) and \(u_2\) the variational solutions in \(\Omega _{\tau _1}\) and \(\Omega _{\tau _2}\), respectively. By the localization principle with respect to time in Lemma 3.3, \(u_2\) is also a variational solution in \(\Omega _{\tau _1}\). Further, \(u_1\) and \(u_2\) coincide in \(\Omega _{\tau _1}\) by the comparison principle in Theorem 3.5. Therefore, a unique global variational solution in the sense of Definition 3.1 can be constructed by taking an increasing sequence of times \((\tau _i)_{i \in \mathbb {N}}\) with \(\lim _{i \rightarrow \infty } \tau _i = \infty \).
Thus we suppose that \(T < \infty \). For \(\varepsilon >0\) we define the Steklov average \(f_\varepsilon :[0,T] \times \mathbb {R}^n \rightarrow \mathbb {R}\) of f by (2.4). A straightforward computation shows that \(\xi \mapsto f_\varepsilon (t,\xi )\) is convex for any \(t \in [0,T]\). Further, for any \(\varepsilon >0\) the derivative of \(f_\varepsilon \) with respect to the time variable is given by
Combining this with (2.1), for any \(L>0\) we have that
Hence, for any \(\varepsilon >0\), the integrand \(f_\varepsilon \) fulfills assumption (4.1). By Theorem 5.2 we conclude that for any \(\varepsilon >0\) there exists a variational solution \(u_\varepsilon \in K^\infty _{u_o}(\Omega _T)\) to the Cauchy–Dirichlet problem associated with \(f_\varepsilon \) in the sense of Definition 1.1 satisfying the bound
Together with the fact that \(u_\varepsilon = u_o\) on \(\partial \Omega \times (0,T)\), this implies in particular that the sequence \((u_\varepsilon )_{\varepsilon >0}\) is bounded in \(L^\infty (\Omega _T)\). Thus, there exists a (not relabelled) subsequence and a limit map \(u \in L^\infty (\Omega _T)\) such that \(u = u_o\) on \(\partial \Omega \times (0,T)\),
and in the limit \(\varepsilon \downarrow 0\) there holds
It remains to show that u is a variational solution to the Cauchy–Dirichlet problem associated with f in the sense of Definition 1.1. To this end, note that \(u_\varepsilon \) satisfies the variational inequality
for any \(\tau \in [0,T] \cap \mathbb {R}\) and any comparison map \(v \in K^\infty _{u_o}(\Omega _\tau )\) with \(\partial _t v \in L^2(\Omega _\tau )\). In the following, we pass to the limit \(\varepsilon \downarrow 0\) in (6.2). In order to treat the left-hand side, we rewrite
By (6.1)\(_3\) and Lemma 2.4 we obtain that
Further, for \(M:= \max \{ Q, \Vert Du_o \Vert _{L^\infty (\Omega ,\mathbb {R}^n)} \}\) we find that
as \(\varepsilon \downarrow 0\) by means of Lemma 2.7. Joining the preceding two estimates yields
Next, by (6.1)\(_1\) we have that
For the second term on the right-hand side of (6.2), by Lemma 2.7 we conclude that
as \(\varepsilon \downarrow 0\). Finally, (6.1)\(_2\) shows that
for a.e. \(\tau \in [0,T]\). Collecting (6.3)–(6.6), we infer that
for a.e. \(\tau \in [0,T]\) and any \(v \in K^\infty _{u_o}(\Omega _\tau )\) with \(\partial _t v \in L^2(\Omega _\tau )\). In particular, this implies that \(u \in C^0([0,T];L^2(\Omega ))\), see Lemma 3.2. Therefore, we have that \(u \in K^\infty _{u_o}(\Omega _T)\) is a variational solution associated with the integrand f in the sense of Definition 1.1. Finally, by the comparison principle in Theorem 3.5, u is unique. This concludes the proof of Theorem 1.2.
7 Continuity in time (Proof of Theorem 1.3)
To prove Theorem 1.3, we begin by verifying that the unique variational solution u to the Cauchy–Dirichlet problem associated with (1.1) and \(u_o\) in \(\Omega _T\) is a weak solution to (1.1) in \(\Omega _T\). To this end, let \(\varphi \in C_0^\infty (\Omega _T)\) be a test function. We want to show that
We set \(v_h:= [u]_h + s[\varphi ]_h\), where in the convolution we use the starting values \(u_o\) and \(\varphi (0) = 0\) for u and \(\varphi \), respectively. Using \(v_h\) as a comparison function in (1.3) and omitting the boundary term at T, we obtain that
Since by (1.4) we have that
it follows from (2.1) that the sequence of mappings \((x,t) \mapsto f(t, Dv_h(x,t))\) has an integrable dominant independent of h. Therefore by the dominated convergence theorem, we conclude that
Further, by integration by parts and the convergence assertions from Lemmas 2.8 and 2.9, we find that
in the limit \(h \downarrow 0\). Thus, letting \(h \downarrow 0\) in (7.2) and dividing by s we deduce that
Finally, observe that by the gradient bound (1.4) and the assumption (1.5), the integrand at the right-hand side of the above inequality is bounded. Thus we may let \(s \rightarrow 0\) to obtain that
The reverse inequality in (7.1) follows by replacing \(\varphi \) by \(-\varphi \).
Consider cylinders of the form
where \((x_0, t_0) \in \overline{\Omega _T}\) and \(r>0\). We show that u satisfies the Poincaré inequality
for all small \(r>0\), where the mean value of u over \(Q_r\) is denoted by
Thus the gradient bound (1.4) together with condition (1.5) yields
for all \(r > 0\). The claim then follows from [9, Theorem 3.1].
To prove (7.3), we first note that since \(\Omega \) is a convex domain, there exist positive constants \(R(\Omega )\) and \(C(\Omega )\) such that for any \(r\in (0,R)\) and \(x_0 \in \overline{\Omega }\), the set \(\Omega \cap B_r(x_0)\) contains a ball of radius \(r/C(\Omega )\). Then we assume that \(Q_r\) with \(r<R\) is given and denote \(B_r:= B_r(x_0)\), \(t_1:= \max (t_0 - r^2, 0)\), \(t_2:= \min (t_0 + r^2, T)\) so that \(Q_r = (B_r \cap \Omega )\times (t_1, t_2)\). We fix a non-negative weight function \(\eta \in C_0^\infty (B_r\cap \Omega )\) such that
For the second assertion, we have used that \(B_{r}\cap \Omega \) contains a ball of size \(r/C(\Omega )\). Since \(B_{r}\cap \Omega \) is convex, the Poincaré inequality
holds for any \(v\in W^{1,2}(B_{r}\cap \Omega )\), see for example [1]. An application of Hölder’s and Minkowski’s inequalities on the above further yields
with a constant \(c=c(n,\Omega )\). We denote the weighted mean of u at time t by
and decompose the left-hand side of (7.3) as follows
To estimate \(I_3\), we apply (7.5) to obtain that
The same estimate holds for \(I_2\) since by Hölder’s inequality we have that
To estimate \(I_1\), let \(\tau _1, \tau _2 \in (t_1, t_2)\) with \(\tau _1 < \tau _2\). As shown above, u is a weak solution to (1.1). That is, abbreviating \(F(x,t):= D_\xi f(t, Du(x,t))\), we find that
Fix \(\delta >0\) and consider
Using the test function \(\varphi (x,t):=\eta \psi _\delta \) in the weak Euler–Lagrange equation yields
Passing to the limit \(\delta \downarrow 0\), the preceding inequality implies that
holds true for almost every \(\tau _1, \tau _2 \in (t_1,t_2)\). In the last inequality, we used that \(\tau _2 - \tau _1 \le t_2 - t_1 \le 2r^2\). Thus
Inequality (7.3) now follows by combining the estimates of \(I_1, I_2\) and \(I_3\).
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Jarkko Siltakoski was funded by the Magnus Ehrnrooth foundation.
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Schätzler, L., Siltakoski, J. The bounded slope condition for parabolic equations with time-dependent integrands. Nonlinear Differ. Equ. Appl. 30, 76 (2023). https://doi.org/10.1007/s00030-023-00876-6
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DOI: https://doi.org/10.1007/s00030-023-00876-6