Abstract
Minima of functionals of the type
with \(\varOmega \subset {\mathbb {R}}^n\), have locally Hölder continuous gradient provided \(1< q < 1+\alpha /n\).
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1 Introduction
In this paper we prove the first Schauder type results for minima of nonuniformly elliptic integrals with nearly linear growth, and under optimal bounds on the rate of nonuniform ellipticity. This is a non-trivial task, as in nonuniformly elliptic problems Schauder type estimates change their nature and cannot be achieved via perturbation methods. This was first shown by the counterexamples in [28, 30]. A relevant model functional we have in mind is
Here, as also in what follows, \(\varOmega \subset {\mathbb {R}}^n\) denotes an open subset, \(n\ge 2\) and \(q>1\); as our results will be local in nature, without loss of generality we assume that \(\varOmega \) is also bounded. The functions \(a(\cdot ), \mathfrak {c}(\cdot )\) are bounded and non-negative with \(\mathfrak {c}(\cdot )\) which is also bounded away from zero; a crucial point here is that \(a(\cdot )\) is instead allowed to vanish. For the definition of local minimizer see (1.15). We are interested in the case \(a(\cdot ), \mathfrak {c}(\cdot )\) are only Hölder continuous, when proving gradient Hölder regularity of minima in fact corresponds to establish a nonlinear version of Schauder estimates (originally also due to Hopf and Caccioppoli). The functional in (1.1) appears to be a combination of the classical nearly linear growth one
treated for when \(\mathfrak {c}(\cdot )\equiv 1\), as was done, for instance, in [33] and by Marcellini & Papi in [62], with a q-power growth term weighted by a non-negative coefficient \(a(\cdot )\). The terminology “nearly linear” accounts for an integrand whose growth in the gradient variable is superlinear, but it is still slower than that of any power type integrand as \(z \mapsto |z|^p\) with \(p>1\). When \(\mathfrak {c}(\cdot )\equiv 1\) the functional 
can be in fact considered as a limiting case (\(p\rightarrow 1\)) of the by now largely studied double phase integral
This last functional is a basic prototype of a nonautonomous one featuring nonstandard polynomial growth conditions and a soft kind of nonuniform ellipticity (see discussion below). Originally introduced by Zhikov [70,71,72] in the setting of Homogenization of strongly anisotropic materials, the functional in (1.3) was first studied in [18, 28, 30] and can be thought as a model for composite media with different hardening exponents p and q. The geometry of the mixture of the two materials is in fact described by the zero set \(\{a(x)\equiv 0\}\) of the coefficient \(a(\cdot )\), where the transition from q-growth to p-growth takes place. In turn, the functional in (1.2) appears for instance in the theory of plasticity with logarithmic hardening, another borderline case between plasticity with power hardening (\(p>1\)) and perfect plasticity (\(p=1\)) [32, 34]. Here, we are mostly interested in the theoretical issues raised by functionals as (1.1) in the context of regularity theory. In fact, 
provides a basic example of nonuniformly and nonautonomous elliptic functional for which the currently available techniques do not allow to prove sharp Schauder type estimates. In particular, although being a limiting case of (1.3), there is no way to adapt the methods available for (1.3) to treat 
, already when \(\mathfrak {c}(\cdot )\equiv 1\). We recall that general nonuniformly elliptic problems have been the object of intensive investigation over the last two decades [7, 9,10,11,12,13,14, 33], starting from the more classical case of minimal surface type functionals [54, 65] and those from the 60 s [45, 46, 48, 64, 68]. The monograph [47] gives a valuable account of the early age of the theory and we recall that nonuniformly elliptic energies have been considered several times in the setting of nonlinear elasticity [15, 29, 31, 57, 58]. As for linear and nearly linear growth conditions in the vectorial case, we mention the recent extensive work of Kristensen and Gmeineder [38,39,40,41]. But let us first recall the situation for (1.3). The bound
guarantees the local Hölder continuity of the gradient of minima of (1.3) [5, 18, 21]. Condition (1.4) is optimal in the sense that its failure generates the existence of minimizers that do not even belong to \(W^{1,q}_{\textrm{loc}}\) [28] and develop singularities on fractals with almost maximal dimension [30]. In (1.4) notice the delicate interaction between ambient dimension, the (p, q)-growth conditions in the gradient and the regularity of coefficient \(a(\cdot )\). Despite the simple form of (1.3), all this already happens in the scalar case. In the autonomous case bounds of the type in (1.4), with no appearance of \(\alpha \), play a key role in the regularity theory of functionals with nonstandard growth as built by Marcellini in his by now classical papers [57, 59,60,61]. The key property of the functional (1.3) is that it is uniformly elliptic in the classical (pointwise) sense. This means that, with \(G(x,z):= |z|^{p}+a(x)|z|^q\), we have
where 
denotes the (pointwise) ellipticity ratio of \(G(x, \cdot )\), i.e.,
Nevertheless, the functional 
exhibits a weaker form of nonuniform ellipticity, detectable via a larger quantity called nonlocal ellipticity ratio [23]. This is defined by
where \(B \subset \varOmega \) is any ball. Indeed, note that
holds whenever \(\{a(x)\equiv 0\}\cap B\) is nonempty, and it is therefore unbounded with respect to |z|. We refer to [23] for a discussion on these quantities; we just note that they do coincide in the autonomous case. It is precisely the occurrence of both (1.5) and (1.6) to imply the regularity of minima of the double phase functional in (1.3) when assuming (1.4). Indeed (1.4) and (1.6), allow to implement a delicate perturbation method based on the fact that minimizers of the frozen integral \(w \mapsto \int G(x_0, Dw)\, \textrm{d}x\) enjoy good decay estimates, which is a consequence of (1.5). A similar scheme, based on the occurrence of pointwise uniform ellipticity (1.5), is the starting point for treating larger classes of problems featuring non-polynomial ellipticity conditions, see for instance the recent interesting papers by Hästo and Ok [43, 44]. These also include integrals of the type
where \(t \mapsto \varPhi _1(t)\) has superlinear growth (these have been treated in [6, 17]). Such schemes completely break down in the case of functionals as 
since the boundedness of 
fails for \(G(x, z)\equiv \mathfrak {c}(x)|z|\log (1+|z|)+a(x)|z|^q\). Indeed, already in the case of the functional in (1.2), which is nonuniformly elliptic in the classical sense (i.e., (1.5) fails), no Lipschitz continuity result for minima is available but when considering differentiablity assumptions on \(\mathfrak {c}(\cdot )\) (see [22]). In other words, the validity of Hopf-Caccioppoli-Schauder estimates is an open issue. We fix the current situation in the following:
Theorem 1.1
(Nearly linear Hopf–Caccioppoli–Schauder) Let \(u\in W^{1,1}_{\textrm{loc}}(\varOmega )\) be a local minimizer of the functional 
in (1.1), with
where \(\alpha , \alpha _0\in (0,1)\) and \(\varLambda \ge 1\). Then Du is locally Hölder continuous in \(\varOmega \) and moreover, for every ball \(B\equiv B_{r}\Subset \varOmega \), \(r\le 1\), the inequality
holds with \(c\equiv c(n,q, \varLambda ,\alpha , \alpha _0,\Vert a \Vert _{C^{0,\alpha }}, \Vert \mathfrak {c} \Vert _{C^{0,\alpha _0}})\) and \(\vartheta \equiv \vartheta (n,q,\alpha , \alpha _0)>0\).
Of course the condition \(q<1+\alpha /n\) in (1.7)\(_1\) is the sharp borderline version of (1.4) (let \(p\rightarrow 1\)), that guarantees gradient Hölder continuity of minima for the classical double phase functional (1.3). Let us only observe that the equality case in (1.4), in comparison to (1.7)\(_1\) assumed here, is typically linked to superlinear growth conditions \(p>1\) in (1.3). The equality in (1.4) is indeed obtained via Gehring type results and it is ultimately again an effect of the uniform ellipticity in the sense of (1.5) (see [5, 20]). Such effects are missing in the nearly linear growth case. Once local gradient Hölder continuity is at hand, in the non-singular/degenerate case we gain
Corollary 1.2
(Improved exponents) Under assumptions (1.7), local minimizers of the functional
are locally \(C^{1,{\tilde{\alpha }}/2}\)-regular in \(\varOmega \), where \({\tilde{\alpha }}:=\min \{\alpha , \alpha _0\}\). In particular, local minimizers of the functional in (1.2) are locally \( C^{1,\alpha _0/2}\)-regular in \(\varOmega \), provided that (1.7)\(_2\) holds.
We do not expect that, at least with the currently available techniques, the exponent \({\tilde{\alpha }}/2\) in Corollary 1.2 can be improved [55, 56]. The loss of 1/2 in the exponent is typical when dealing with functionals with subquadratic growth.
1.1 General Statements
Theorem 1.1 and Corollary 1.2 are special cases of a class of results covering general functionals of the type
with \(F(\cdot )\) being arbitrarily close to have linear growth and \(\texttt {s} \in [0,1]\). This time we are thinking of integrands of the type
where \(\mathfrak {c}(\cdot )\) is as in (1.7)\(_2\). For situations like this, we prepare a more technical theorem. We consider continuous integrands \(F:\varOmega \times {\mathbb {R}}^n\rightarrow [0, \infty )\) such that \(z \mapsto F(x, z) \in C^{2}({\mathbb {R}}^n)\) for every \(x\in \varOmega \) and satisfying
for every choice of \(x,y\in \varOmega \), \(z, \xi \in {\mathbb {R}}^n\), with \(\alpha _0\in (0,1)\), \(1\le \mu <3/2\) and \(0 < \nu \le 1 \le L\) being fixed constants. Here \(g:[0, \infty ) \rightarrow [1, \infty )\) is a non-decreasing, concave and unbounded function such that \( t\mapsto tg(t) \) is convex. Moreover, we assume that for every \(\varepsilon >0\) there exists a constant \(c_{g}(\varepsilon )\) such that
The above assumptions are sufficient to get Lipschitz continuity of minimizers. In order to get gradient local Hölder continuity we still need a technical one (see also Remark 1 below):
For functionals as in (1.9) we have
Theorem 1.3
Let \(u\in W^{1,1}_{\textrm{loc}}(\varOmega )\) be a local minimizer of the functional in (1.9) under assumptions (1.7)\(_1\) and (1.11)–(1.12). There exists \(\mu _{\text {max}}\in (1,3/2)\), depending only \(n,q, \alpha , \alpha _0\), such that, if \(1\le \mu < \mu _{\text {max}}\), then
holds whenever \(B\equiv B_{r}\Subset \varOmega \), \(r\le 1\), where \(c\equiv c(\texttt {data} )\) and \(\vartheta \equiv \vartheta (n,q, \alpha , \alpha _0)>0\). Moreover, assuming also (1.13) implies that Du is locally Hölder continuous is \(\varOmega \). Finally, if in addition to all of the above we also assume that \(\texttt {s} >0\) and that \(\partial _{zz}F(\cdot )\) is continuous, then \(u \in C^{1,{\tilde{\alpha }}/2}_{\textrm{loc}}(\varOmega )\), with \({\tilde{\alpha }} = \min \{\alpha , \alpha _0\}\).
The notion of local minimizer we are using in this paper is classical and prescribes that a function \(u \in W^{1,1}_{\textrm{loc}}(\varOmega )\) is a local minimizer of 
if, for every ball \(B\Subset \varOmega \), it happens that
This implies, in particular, that 
is finite whenever \(\varOmega _0\Subset \varOmega \) is an open subset. In Theorem 1.3, we are using the shorthand notation
Which we adopt for the rest of the paper. For further notation we refer to Section 2 below. We just remark that, in order to simplify the reading, in the following we shall still denote by \(c \equiv c (\texttt {data} )\) a constant actually depending only on a subset of the parameters listed in (1.16).
Theorem 1.3 has a technical nature, especially as far as the size of \(\mu _{\text {max}}\) is concerned, which is not allowed to be too far from one (a natural limitation in view of the condition \(q < 1+\alpha /n\)). This is anyway sufficient to cover the model examples considered here, including (1.10); see Section 5. We note that in the autonomous, nearly linear case, the literature reports several interesting results for functionals \(w \mapsto \int F(Dw)\, \textrm{d}x\), satisfying (1.11)\(_{1,2}\) with restrictions as \(1\le \mu < 1+\texttt {o}(n)\), \(\texttt {o}(n) \rightarrow 0\) when \(n\rightarrow \infty \). These bounds are in fact of the type we also need to impose when selecting \(\mu _{\text {max}}\) (this can be checked by tracking the constant dependence in the proofs). Starting by [33] conditions (1.11)\(_2\) have been used in several papers devoted to functionals with linear or nearly linear growth; see for instance [12, 13, 41] and related references.
Remark 1
The technical assumption (1.13) is automatically implied, for a different constant L, by \(\partial _{z}F(x, 0_{{\mathbb {R}}^n})=0_{{\mathbb {R}}^n}\); this follows using the second inequality in (1.11)\(_2\). In turn, when \(F(\cdot )\) is independent of x we can always assume that \(\partial _{z}F(0_{{\mathbb {R}}^n})=0_{{\mathbb {R}}^n}\). Indeed, we can switch from F(z) to \(F(z)-\langle \partial _zF(0_{{\mathbb {R}}^n}), z\rangle \) observing that such a replacement in the integrand in (1.9) does not change the set of local minimizers. Assumption (1.13) is anyway verified by the model cases we are considering and in fact it deals with the behaviour of \(F(\cdot )\) at the origin, while in nonuniformly elliptic equations the main problems usually come from the behaviour at infinity.
1.2 Techniques
The techniques needed for Theorem 1.3 are entirely different from those used in the case of classical superlinear double phase functionals in (1.3) [5, 18] and for pointwise uniformly elliptic problems with nonstandard growth conditions [43, 44]. This is essentially due to the failure of uniform ellipticity (1.5) for functionals as 
and 
. On the other hand, the techniques based on De Giorgi and Moser’s methods used in the literature to deal with nearly linear growth functionals [12, 14, 32, 33] are not viable here. In fact, as in our case coefficients are Hölder continuous, it is impossible to differentiate the Euler-Lagrange equation to get gradient regularity. This is in fact the classical obstruction to direct regularity proofs one encounters when proving Schauder estimates, eventually leading to the use of perturbation methods. In turn, perturbation methods are again not working in the nonuniformly elliptic case as the available a priori estimates are not homogeneous and iteration schemes fail. Our approach is new to the context and relies on a number of ingredients, partially already introduced and exploited in [24]. Most notably, we make a delicate use of nonlinear potential theoretic methods allowing to work under the sharp bound
We employ a rather complex scheme of renormalized fractional Caccioppoli inequalities on level sets (5.36). These are aimed at replacing the standard Bernstein technique, and encode the essential data of the problem via a wide range of parameters; see (5.37)–(5.39). Bernstein technique is traditionally based on the fact that functions as \(|Du|^\gamma \) are subsolutions to linear elliptic equations for certain positive values of \(\gamma \). This follows differentiating the original equation, something that is impossible in our setting. We therefore use a fractional and anisotropic version of the method according to which certain nonlinear functions of the gradient, namely
belong to suitable fractional De Giorgi’s classes, i.e., they satisfy inequalities (5.36). The term renormalized accounts for the fact that inequalities (5.36) are homogeneous with respect to \(E(\cdot ,Du)\), despite the functional 
is not. This costs the price of getting multiplicative constants depending on \(\Vert Du\Vert _{L^{\infty }}\). Such constants must be carefully kept under control all over the proof and reabsorbed at the very end. Notice that inequalities (5.36) are obtained via a dyadic/atomic decomposition technique, finding its roots in [52], that resembles the one used for Besov spaces [2, 67] in the setting of Littlewood-Paley theory; see Section 5.6 and Remark 3. Iterating inequalities (5.36) via a potential theoretic version of De Giorgi’s iteration (Lemma 2.5) then leads to establish local Lipschitz bounds in terms of the nonlinear potentials defined in (2.11); see Proposition 5.5. A crucial point is that the fractional nature of the inequalities (5.36) allows to sharply quantify how the rate of Hölder continuity of coefficients interacts with the growth of the terms they stick to. Using this last very fact and combining the various parameters in (5.37)–(5.39) leads to determine the optimal nonlinear potentials allowing to prove Lipschitz continuity of local minimizers under the sharp bound in (1.17), but no matter how small the exponent \(\alpha _0\) is. Throughout the whole process, the structure of the functional 
needs to be carefully exploited at every point. This reflects in the peculiar choice of (1.18) as the leading quantity to iterate. Lipschitz regularity is as usual the focal point in nonuniformly elliptic problems, since once Du is (locally) bounded condition (1.5) gets automatically satisfied when \(z\equiv Du\). Indeed at this stage gradient Hölder continuity can be achieved by exploiting and extending some hidden facts from more classical regularity theory; see Section 5.10. In particular, we shall use some of the iteration schemes employed in the proof of nonlinear potential estimates for singular parabolic equations presented in [53]. The arguments developed for this point are general and can be used in other settings; see Remark 4. We finally observe that in this paper we preferred to concentrate on model cases in order to highlight the main ideas. Nevertheless we believe that the approach included here can be exported to a variety of different settings where it might yield results that are either sharp or unachievable otherwise.
2 Preliminaries
2.1 Notation
In the following we denote by c a general constant larger than 1. Different occurrences from line to line will be still denoted by c. Special occurrences will be denoted by \(c_*, {\tilde{c}}\) or likewise. Relevant dependencies on parameters will be as usual emphasized by putting them in parentheses. We denote by \( B_{r}(x_0):= \{x \in {\mathbb {R}}^n: |x-x_0|< r\}\) the open ball with center \(x_0\) and radius \(r>0\); we omit denoting the center when it is not necessary, i.e., \(B \equiv B_{r} \equiv B_{r}(x_0)\); this especially happens when various balls in the same context share the same center. We also denote
when the ball in question is centred at the origin. Finally, with B being a given ball with radius r and \(\gamma \) being a positive number, we denote by \(\gamma B\) the concentric ball with radius \(\gamma r\) and by \(B/\gamma \equiv (1/\gamma )B\). Moreover, \(Q_{\text {inn}}\equiv Q_{\text {inn}}(B)\) denotes the inner hypercube of B, i.e., the largest hypercube, with sides parallel to the coordinate axes and concentric to B, that is contained in B, \(Q_{\text {inn}}(B)\subset B\). The sidelength of \(Q_{\text {inn}}(B)\) equals \(2r/\sqrt{n}\). The vector valued version of function spaces like \(L^p(\varOmega ), W^{1,p}(\varOmega )\), i.e., when the maps considered take values in \({\mathbb {R}}^k\), \(k\in {\mathbb {N}}\), will be denoted by \(L^p(\varOmega ;{\mathbb {R}}^k), W^{1,p}(\varOmega ;{\mathbb {R}}^k)\); when no ambiguity will arise we shall still denote \(L^p(\varOmega )\equiv L^p(\varOmega ;{\mathbb {R}}^k)\) and so forth. For the rest of the paper we shall keep the notation
for \(z\in {\mathbb {R}}^{k}\), \(k\ge 1\) and \(\omega \in [0,1]\). In particular, when \(k=1\), we have a function defined on \({\mathbb {R}}\). With 
being a measurable subset with bounded positive measure 
, and with 
, \(k\ge 1\), being an integrable map, we denote that
Moreover, given a (scalar) function f and a number \(\kappa \in {\mathbb {R}}\), we denote that
Finally, whenever 
and 
is any set, we define
As usual, with \(\beta \in (0,1]\) we denote that
2.2 Auxiliary Results
We shall use the vector fields \(V_{\omega ,p}:{\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n}\), defined by
whenever \(z \in {\mathbb {R}}^{n}\) (compare [42, (2.1)] for such extended range of p). They satisfy
that holds for every \(z_1, z_2 \in {\mathbb {R}}^n\), \(\omega \in [0,1]\) and \(p>0\); for this we refer to [42, Lemma 2.1]. We shall also use the following, trivial
that holds for every \(\gamma \ge 0\) and whenever \(|z_1|+|z_2|>0\). When restricting from above the range of \(\gamma \), the inequality becomes double-sided, i.e.
that instead holds for every \(\gamma <1\); see [42, 36, Section 2]. With \(B \subset \varOmega \) being a ball and \(\sigma \in (0,1)\), we let
and define
where \(z_{1},z_{2}\in {\mathbb {R}}^{n}\), \(x\in \varOmega \). A consequence of the assumptions of (1.11) is the following:
Lemma 2.1
Let \(F:\varOmega \times {\mathbb {R}}^n\rightarrow [0, \infty )\) be as in (1.11). Then \(|\partial _z F(x,z)| \le cg(|z|)\) holds for every \((x,z)\in \varOmega \), where c depends on L and the constant \(c_{g}(1)\) defined in (1.12).
Proof
This is a version of [59, Lemma 2.1] and the proof mimics that of Marcellini. More specifically, there we take \(|h|=1 +|z|\) and, after using (1.11)\(_1\) as in [59], we estimate \(g(|z\pm he_i|)\le g(|z|+h)\le g(|z|)+g(h)\le 2\,g(|z|)+g(1) \le cg(|z|)\) (here \(\{e_k\}\) denotes the standard basis of \({\mathbb {R}}^n\)). Here we have used that \(g(\cdot )\ge 1\) is non-decreasing and sublinear (since it is concave and non-negative) and finally (1.12) to get \(g(1)\le c_g(1)\le g (|z|)\). \(\square \)
Finally, a classical iteration lemma [37, Lemma 6.1].
Lemma 2.2
Let \(h:[t,s]\rightarrow {\mathbb {R}}\) be a non-negative and bounded function, and let \(a,b, \gamma \) be non-negative numbers. Assume that the inequality \( h(\tau _1)\le (1/2) h(\tau _2)+(\tau _2-\tau _1)^{-\gamma }a+b, \) holds whenever \(t\le \tau _1<\tau _2\le s\). Then \( h(t)\le c( \gamma )[a(s-t)^{-\gamma }+b] \), holds too.
2.3 Fractional Spaces
The finite difference operator \(\tau _{h}:L^1(\varOmega ;{\mathbb {R}}^{k}) \rightarrow L^{1}(\varOmega _{|h|};{\mathbb {R}}^k)\) is defined as \(\tau _{h}w(x):=w(x+h)-w(x)\) for \(x\in \varOmega _{|h|}\), for every \(\tau \in L^1(\varOmega )\) and where \(\varOmega _{|h|}:=\{x \in \varOmega \,: \, \,\textrm{dist}(x, \partial \varOmega ) > |h|\}\). With \(\beta \in (0,1)\), \(s \in [1, \infty )\), \(k \in {\mathbb {N}}\), \(n \ge 2\), and with \(\varOmega \subset {\mathbb {R}}^n\) being an open subset, the space \(W^{\beta ,s}(\varOmega ;{\mathbb {R}}^k )\) is defined as the set of maps \(w :\varOmega \rightarrow {\mathbb {R}}^k\) such that
When considering regular domains, as for instance the ball 
(this is the only case we are going to consider here in this respect), the embedding inequality
holds provided \(s\ge 1, \beta \in (0,1)\), \(k\in \mathbb {N}\) and \(s\beta <n\). For this and basic results concerning Fractional Sobolev spaces we refer to [27] and related references. A characterization of fractional Sobolev spaces via finite difference is contained in the following lemma, that can be retrieved for instance from [22]:
Lemma 2.3
Let \(B_{\varrho } \Subset B_{r}\subset {\mathbb {R}}^n\) be concentric balls with \(r\le 1\), \(w\in L^{s}(B_{r};{\mathbb {R}}^k)\), \(s\ge 1\) and assume that, for \(\alpha _* \in (0,1]\), \(S\ge 1\), if holds that
Then
holds for every \(\beta < \alpha _*\), with \(c\equiv c(n,s)\).
2.4 Nonlinear Potentials
We shall use a family of nonlinear potentials that, in their essence, goes back to the fundamental work of Havin and Mazya [63]. We have recently used such potentials in the setting of nonuniformly elliptic problems in [8, 19, 23,24,25] and in fact we shall in particular use [24, Section 4] as a main source of tools in this respect. With \(t,\sigma >0\), \(m, \theta \ge 0\) being fixed parameters, and \(f\in L^{1}(B_{r}(x_{0}))\) being such that \(|f|^{m} \in L^{1}(B_{r}(x_0))\), where \(B_{r}(x_0)\subset {\mathbb {R}}^n\) is a ball, the nonlinear Havin–Mazya–Wolff type potential \(\textbf{P}_{t,\sigma }^{m,\theta }(f;\cdot )\) is defined by
Suitable integrability properties of f guarantee that \(\textbf{P}_{t,\sigma }^{m,\theta }(f; \cdot )\) is bounded, as stated in
Lemma 2.4
Let \(t,\sigma ,\theta >0\) be numbers such that
Let \(B_{\varrho }\subset B_{\varrho +r_{0}}\subset {\mathbb {R}}^{n}\) be concentric balls with \(\varrho , r_0\le 1\) and \(f\in L^{1}(B_{\varrho +r_{0}})\) be a function such that \(|f|^{m}\in L^{1}(B_{\varrho +r_{0}})\), for some \(m>0\). Then
and with \(c\equiv c(n,t,\sigma ,m,\theta , \gamma )\).
Note that in (2.13) we are not requiring that \(\gamma \ge 1\) and any value \(\gamma >0\) is allowed. Next lemma is a version of [24, Lemma 2.3] and takes information from various results from Nonlinear Potential Theory starting by the seminal paper [49]. It is essentially a nonlinear potential theoretic version of De Giorgi’s iteration. Notice that the very crucial point in the next statement is the tracking, in the various inequalities, of the explicit dependence on the constants \(M_0, M_i\).
Lemma 2.5
Let \(B_{r_{0}}(x_{0})\subset {\mathbb {R}}^{n}\) be a ball, \(n\ge 2\), and, for \(j \in \{1, 2,3\}\), consider functions \(f_j\), \( |f_j|^{m_j} \in L^1(B_{2r_0}(x_{0}))\), and constants \(\chi >1\), \(\sigma _j, m_j, \theta _j>0\) and \(c_*,M_0 >0\), \(\kappa _0, M_j\ge 0\). Assume that \(w \in L^2(B_{r_0}(x_0))\) is such that for all \(\kappa \ge \kappa _{0}\), and for every ball \(B_{\varrho }(x_{0})\subseteq B_{r_{0}}(x_{0})\), the inequality
holds (recall the notation in (2.2)). If \(x_{0}\) is a Lebesgue point of w, then
holds with \(c\equiv c(n,\chi ,\sigma _j,\theta _j,c_{*})\).
3 Auxiliary Integrands and their Eigenvalues
With reference to a general functional of the type 
in (1.9), considered under the assumptions of Theorem 1.3 (in particular implying that \(q<3/2\)), we define the integrand
for \(z \in {\mathbb {R}}^n\), \(x\in \varOmega \). We fix an arbitrary ball \(B\subset \varOmega \) centred at \(x_{\textrm{c}}\) and, with \(\omega \in [0,1]\) and \(\sigma \in (0,1)\), recalling (2.1) and (2.7), we define
It follows that
Note that abbreviations as \(H_{\omega ,\sigma ,\text {i}}(z)\equiv H_{\omega ,\sigma ,\text {i}}(z;B)\), used in (3.2), will take place whenever there will be no ambiguity concerning the identity of the ball B in question. A similar shortened notation will be adopted for the case of further quantities depending on a single ball B. We next introduce two functions, \(\lambda _{\omega ,\sigma },\varLambda _{\omega ,\sigma }:\varOmega \times [0, \infty )\rightarrow [0, \infty )\), bound to describe the behaviour of the eigenvalues of \(\partial _{zz}H_{\omega ,\sigma }(\cdot )\). These are (recall that \(q, \mu <3/2\))
It then follows that
hold for all \(x,y\in \varOmega \), \(z,z_{1},z_{2},\xi \in {\mathbb {R}}^{n}\), where \(c \equiv c (\texttt {data} )\) (recall the definition of 
in (2.8) and (2.3)). Notice that (3.5)\(_{1,2}\) directly follow from (1.11)\(_{1,2}\), respectively, and (1.12) used with \(\varepsilon =q-1\). The inequality in (3.5)\(_{3}\) is a straightforward consequence of (2.4)–(2.5) and (3.5)\(_{2}\) (see Remark 2 below). Finally, (3.5)\(_4\) is a consequence of (1.11)\(_{3}\). When \(\omega >0\), by (1.12) and (3.5)\(_2\) the integrand \(H_{\omega ,\sigma }(\cdot )\) also satisfies, for all \(x,y \in \varOmega \), \(|x-y|\le 1\) and \(z,\xi \in {\mathbb {R}}^{n}\)
with \({\tilde{\alpha }}=\min \{\alpha , \alpha _0\}\) and \(c\equiv c(\texttt {data} , \omega ).\) Note that, on the contrary of the previous displays, the constant c in (3.6) depends on \(\omega \) and that for (3.6)\(_2\) we also need Lemma 2.1 and (3.5)\(_{2}\). As for \(H_{\omega ,\sigma ,\text {i}}(\cdot )\), by defining
we have that
holds whenever \(z,z_{1},z_{2},\xi \in {\mathbb {R}}^{n}\), with \(c\equiv c(\texttt {data} )\); for (3.8)\(_2\) we have used Lemma 2.1. In particular, it holds that
(recall that \(g(\cdot )\ge 1\)). As in (3.6), when \(\omega >0\) the function \(H_{\omega ,\sigma ,\text {i}}(\cdot )\) satisfies
for all \(z,\xi \in {\mathbb {R}}^{n}\), with \(c\equiv c(\texttt {data} , \omega ).\) Two quantities that will have an important role in the forthcoming computations are, for \(x\in \varOmega \) and \(t\ge 0\),
It follows that (recall that \(\mu <2\))
and this leads us to define that
We record a few basic properties of these functions in the following:
Lemma 3.1
The following holds about the functions in (3.11), whenever \(B\subseteq \varOmega \) is a ball:
-
For every \(s,t\in [0,\infty )\)
$$\begin{aligned}&|E_{\omega ,\sigma ,\text {i}}(s)-E_{\omega ,\sigma ,\text {i}}(t)|\nonumber \\&\quad \le \left[ (s^2+t^2+1)^{(1-\mu )/2}+a_{\sigma , \text {i}}(B)(s^2+t^2+\omega ^2)^{(q-1)/2}\right] |s-t|. \end{aligned}$$(3.14) -
For all \(x\in B\)
$$\begin{aligned} |E_{\omega ,\sigma }(x,t)-E_{\omega ,\sigma , \text {i}}(t)|\le (1-1/q) |a(x)-a_{\text {i}}(B)|\left( [\ell _{\omega }(t)]^{q}-\omega ^q\right) . \end{aligned}$$(3.15) -
There exists constant \(T\equiv T(\mu )\ge 1\) such that
$$\begin{aligned} t\le 2 [E_{\omega ,\sigma }(x,t)]^{\frac{1}{2-\mu }} \quad \text{ and }\quad t\le 2[E_{\omega ,\sigma ,\text {i}}(t)]^{\frac{1}{2-\mu }} \end{aligned}$$(3.16)hold for all \(x\in \varOmega \), \(t\ge T\).
-
There exists a constant \(c\equiv c(\mu , q,\nu )\) such that
$$\begin{aligned} {\left\{ \begin{array}{ll} |z|+{\tilde{E}}_{\omega ,\sigma }(x,|z|)\le c [H_{\omega ,\sigma }(x,z)+1]\\ |z|+{\tilde{E}}_{\omega ,\sigma , \text {i}}(|z|) \le c [H_{\omega ,\sigma ,\text {i}}(z)+1] \end{array}\right. } \end{aligned}$$(3.17)hold for every \(x\in \varOmega \) and \(z\in {\mathbb {R}}^n\).
Proof
The only inequality deserving some comments is (3.14), the other being direct consequences of the definitions in (3.12)–(3.13). Notice that
with \(c\equiv c (q, \mu )\). Using this last inequality we have
so that (3.14) follows applying (2.6) with \(\gamma \equiv \mu -1\) and \(\gamma \equiv 1-q\). \(\square \)
Remark 2
The proof of (3.5)\(_{3}\) is rather standard, but we report it for completeness as the exponents involved here are not the usual ones. Using, in turn, (2.4) (with \(p=2-\mu , q\)) and (2.5) (with \(\gamma = \mu , 2-q\)), and finally the first inequality in (3.5)\(_{2}\), we have
4 Lipschitz Estimates with Small Anisotropicity
Let \(B_{\tau }\equiv B_{\tau }(x_{\textrm{c}})\Subset \varOmega \) be a ball with \(\tau \le 1\) and let \(u_{0}\in W^{1,\infty }(B_{\tau })\). With \(\omega \in (0,1]\), we define \(v_{\omega ,\sigma }\in u_{0}+W^{1,q}_{0}(B_{\tau })\) as the unique solution to the Dirichlet problem
where, according to the notation in (3.2)\(_2\), it is \(H_{\omega ,\sigma ,\text {i}}(z)\equiv H_{\omega ,\sigma ,\text {i}}(z;B_{\tau })\). In particular, we have
Existence, and uniqueness, follow by Direct Methods of the Calculus of Variations and the Euler-Lagrange equation
holds for all \(\varphi \in W^{1,q}_{0}(B_{\tau })\) by (3.10). Again (3.10) and standard regularity theory [37, Chapter 8], [55, 56] imply
Proposition 4.1
With \(v_{\omega ,\sigma }\in u_{0}+W^{1,q}_{0}(B_{\tau })\) as in (4.1), for every integer \(n\ge 2\) there exists a continuous and non-decreasing function \(\mu _n :(0,1) \rightarrow (1, 3/2)\) such that, for every \(\delta \in (0,1)\), if (1.11) holds with \(1\le \mu < \mu _n(\delta )\), then
holds too, with \(c\equiv c(\texttt {data} , \delta )\ge 1\), where \({\mathbb {H}}_{\omega ,\sigma ,\text {i}}(\cdot )\) has been defined in (3.2)\(_3\). Moreover, whenever M is a number such that
where \(B\Subset B_{\tau }\) is a ball (not necessarily concentric to \(B_{\tau }\)), the Caccioppoli type inequality
holds whenever \(\kappa \ge 0\), again with \(c\equiv c(\texttt {data} , \delta )\). Here, according to the notation in (3.11)\(_2\), it is \(E_{\omega ,\sigma ,\text {i}}(|Dv_{\omega ,\sigma }|)\equiv E_{\omega ,\sigma ,\text {i}}(|Dv_{\omega ,\sigma }|;B_{\tau })\).
Proof
The final shape of \(\mu _n>1\) will be determined in the course of the proof via successive choices of \(\mu \) (to be taken closer and closer to one, depending on \(\delta \)). All the constants in the forthcoming estimates will be independent of \(\omega ,\sigma \), so we shall omit indicating the dependence on such parameters, simply denoting \(v \equiv v_{\omega ,\sigma }\), \(H_{\text {i}}\equiv H_{\omega ,\sigma ,\text {i}}\), \({\mathbb {H}}_{\text {i}}\equiv {\mathbb {H}}_{\omega ,\sigma ,\text {i}}\), \(E_{\text {i}}\equiv E_{\omega ,\sigma ,\text {i}}\), and so on. Properties (4.4) allow to differentiate (4.3), i.e., replacing \(\varphi \) by \(D_s\varphi \) for \(s \in \{1, \ldots , n\}\) and integrate by parts. Summing the resulting equations over \(s \in \{1, \ldots , n\}\), we get
that, again thanks to (4.4), holds for all \(\varphi \in W^{1,2}(B_{\tau })\) such that \(\,\textrm{supp}\, \varphi \Subset B_{\tau }\). Let \(B\Subset B_{\tau }\) be a ball, \(\kappa \ge 0\) be any non-negative number and \(\eta \in C^{1}_{c}(B_{\tau })\) be a cut-off function such that \(\mathbb {1}_{3B/4}\le \eta \le \mathbb {1}_{5B/6}\) and \(|D\eta |\lesssim |B|^{-1/n}\). By (4.4) the functions \(\varphi \equiv \varphi _s:=\eta ^{2}(E_{\text {i}}(|Dv|)-\kappa )_{+}D_{s}v\) are admissible in (4.8), therefore via (3.7)–(3.8) and Young’s inequality we obtain
with \(c\equiv c (\texttt {data} )\). See [8, Lemmas 4.5\(-\)4.6] for more details and a similar inequality. Using (3.9) and a few elementary manipulations, we again find
and taking M as in (4.6) we arrive at
again with \(c\equiv c(\texttt {data} )\). In turn, using (1.12) with \(\varepsilon >0\) and recalling that \(M\ge 1\), we find
again with \(c\equiv c (\texttt {data} , \varepsilon )\). This gives (4.7) recalling that \(\eta \ge 1\) on 3B/4 and provided \(\mu \equiv \mu (\delta ) \ge 1\) and \(\varepsilon \equiv \varepsilon (\delta )>0\) are such that
We pass to the proof of (4.5). With T being the constant in (3.16), we can assume without loss of generality that
holds, otherwise (4.5) follows trivially. By (4.10) and Sobolev embedding theorem we obtain
where \(2\chi \equiv 2\chi (n)>2\) is the exponent coming from the Sobolev embedding exponent, and \(c\equiv c (\texttt {data} , \varepsilon )\). We now want to apply Lemma 2.5 to \(w\equiv E_{\text {i}}(|Dv|)\) using (4.13) to satisfy (2.14) (here all the terms involving the functions \(f_j\) in (2.14) are not present). We fix parameters \(3\tau /4\le \tau _{1}<\tau _{2}\le 5\tau /6\) and related balls
this time all concentric to the initial ball \(B_{\tau }\equiv B_{\tau }(x_{\textrm{c}})\). Take \(x_{0}\in B_{\tau _{1}}\) arbitrary and set \(r_{0}:=(\tau _{2}-\tau _{1})/8\), so that \( B_{r_{0}}(x_0) \subset B_{\tau _2}\), choose \(M\equiv 2\Vert Dv \Vert _{L^{\infty }(B_{\tau _2})}\) and take \(B\equiv B_{\varrho }(x_0)\subset B_{r_0}(x_0)\) in (4.13). This yields
where \(c\equiv c(\texttt {data} , \varepsilon )\). Note that the above choice of M fits (4.6) via (4.12). By (4.14) we are able to apply Lemma 2.5 with \(\kappa _{0}=0\), \(M_0:= M^{\mu -1+\varepsilon } \) and \(M_1=M_2=M_3\equiv 0\), to get
with \(c\equiv c(n,q)\). Being \(x_{0}\) arbitrary in \(B_{\tau _{1}}\), (4.15) implies
and we have used the actual definition of M. Recalling (4.12), we continue to estimate as
for \(c\equiv c(\texttt {data} , \varepsilon )\). By choosing \(\varepsilon >0\) and \(\mu > 1\) such that
holds (note that this function is increasing in both \(\mu \) and \(\varepsilon \) and that \( \gamma _*(1,0)=1/2\)), we can apply Young’s inequality in (4.16), thereby obtaining that
where
Notice that \(\tilde{\gamma }(1, 0)=1\) and that this is an increasing function of its arguments, therefore we find \(\varepsilon >0\) and \(\mu > 1\) such that
is satisfied in addition to (4.17). Lemma 2.2, applied with
which is always finite by (4.4)\(_2\), now gives
so that, recalling (4.2), we arrive at
Setting \(\tilde{E}_{0}(t):=t+a_{\sigma , \text {i}}(B)t^{q}\) for \(t\ge 0\), we find, also thanks to (3.16), that
where \(c\equiv c (\nu , q)\). Recalling (1.11)\(_1\), the definitions in (3.2) imply
so that inequality (4.19) together with (3.3) and (4.20) now gives
with \(c\equiv c(\texttt {data} , \delta )\), provided we further take \(\mu > 1\) such that
Note that in the above inequality we have incorporated the dependence on \(\varepsilon \) in the dependence on \(\delta \) as this last quantity influences the choice of \(\varepsilon \) via (4.17) and (4.18). Using the very definition (3.2)\(_3\) we then continue to estimate
and, by again using (1.12) with \(\varepsilon =\delta /2\), we conclude with
with \(c\equiv c (\texttt {data} , \delta )\). Since \(\tilde{E}_{0}(\cdot )\) is monotone increasing, convex and such that \(\tilde{E}_{0}(0)=0\), we deduce that its inverse \(\tilde{E}_{0}^{-1}(\cdot )\) is increasing, concave and \(\tilde{E}_{0}^{-1}(0)=0\), therefore it is subadditive and, for any given constant \(c_{*}\ge 0\) it is \(\tilde{E}_{0}^{-1}(c_{*}t)\le (c_{*}+1)\tilde{E}_{0}^{-1}(t)\), see for instance [21, Remark 10 in Section 4]. Using this last property, we can apply \(\tilde{E}_{0}^{-1}(\cdot )\) to both sides of (4.22) in order to conclude with (4.5). Finally, the choice of the function \(\mu _n(\delta )\) mentioned in the statement, that we can always take such that \(\mu _n(\delta )<3/2\), comes by the choices made in (4.11), (4.17), (4.18) and (4.21) and a standard continuity argument (for this recall that both \(\gamma _*(\cdot )\) and \(\tilde{\gamma }(\cdot )\) are increasing functions of their arguments and that \(\chi \) depends on n). Notice that by the above construction it follows that \(\mu _n(0_+)=1\). \(\square \)
5 Proof of Theorem 1.3
The proof will take eleven different steps, distributed along Sections 5.1–5.11 below. We shall concentrate on the singular case \(\texttt {s} =0\), then giving remarks on how to deal with the (actually simpler) case \(\texttt {s} >0\) at the very end of Section 5.9. We shall start assuming that (1.11)–(1.12) for some \(\mu \in [1, 3/2)\). We shall make further restrictions on the size of \(\mu \) in the course of the proof until we finally come to determine the value \(\mu _{\text {max}}\) mentioned in the statement of Theorem 1.3.
5.1 Absence of Lavrentiev Phenomenon
We have the following approximation-in-energy result that actually implies the absence of Lavrentiev phenomenon:
Lemma 5.1
With \(H(\cdot )\) defined in (3.1), assume (1.7)\(_1\) and that \(F(\cdot )\) satisfies (1.11)\(_1\) where \(g(\cdot )\) is as in Theorem 1.3 (described after (1.11)). Let \(w\in W^{1,1}_{\textrm{loc}}(\varOmega )\) be any function such that \(H(\cdot , Dw)\in L^1_{\textrm{loc}}(\varOmega )\). For every ball \(B \Subset \varOmega \) there exists a sequence \(\{w_{\varepsilon }\}\subset W^{1,\infty }(B)\) such that \(w_{\varepsilon }\rightarrow w\) in \(W^{1,1}(B)\) and \(H(\cdot ,Dw_{\varepsilon })\rightarrow H(\cdot ,Dw) \) in \(L^{1}(B)\).
The proof follows [22, Section 5] almost verbatim (see also [28, Lemma 13]). Notice that this is essentially the only point where the assumed convexity of \(t \mapsto tg(t)\) is used (this implies that \(z \mapsto |z|g(|z|)\) is convex as \(g(\cdot )\) is also non-decreasing). For more related results on the absence of Lavrentiev phenomenon we refer to the recent papers [1, 3, 4, 16, 50, 51] and related references.
5.2 Auxiliary Dirichlet Problems and Convergence
In the following \(u\in W^{1,1}_{\textrm{loc}}(\varOmega )\) denotes a local minimizer of 
, as in Theorem 1.3. By \(\omega , \varepsilon \equiv \{\omega \}, \{\varepsilon \} \equiv \{\omega _k\}_k, \{\varepsilon _k\}_k,\) we denote two decreasing sequences of positive numbers such that \(\omega , \varepsilon \rightarrow 0\), and \(\varepsilon ,\omega \le 1\); we will several times extract subsequences and these will still be denoted by \(\omega , \varepsilon \) (for this reason we drop the pendice k). We denote by \(\texttt {o}(\varepsilon )\) a quantity such that \(\texttt {o}(\varepsilon )\rightarrow 0\) as \(\varepsilon \rightarrow 0\). Similarly, we denote by \(\texttt {o}_{\varepsilon }(\omega )\) a quantity, depending both on \(\varepsilon \) and \(\omega \), such that \(\texttt {o}_{\varepsilon }(\omega )\rightarrow 0\) as \(\omega \rightarrow 0\) for each fixed \(\varepsilon \). The exact value of such quantities might change on different occurences and only the aforementioned asymptotic properties will matter. Let \(B_{r}\Subset \varOmega \) be a ball with \(0< r \le 1\); by Lemma 5.1, there exists a sequence \(\{\tilde{u}_{\varepsilon }\}\in W^{1, \infty }(B_{r})\) so that
We define the sequence
(uniformly with respect to \(\omega \in (0,1]\)). Then we consider \(u_{\omega ,\varepsilon }\in \tilde{u}_{\varepsilon }+W^{1,q}_{0}(B_{r})\) as the unique solution to the Dirichlet problem
where
Recall that the integrand \(H_{\omega ,\sigma _{\varepsilon }}(\cdot )\) has been defined in (3.2), with \(\sigma \equiv \sigma _{\varepsilon }\). The solvability of (5.3) follows by Direct Methods and standard convexity arguments. By (3.6), we can apply the by now classical regularity theory contained in [37, Chapter 8] and [55, 56], therefore
Mean value theorem and (5.2) imply
Similarly, noting that \(|z|^{q-1} \le |z| +1 \le c F(x,z) +c\) by (1.11)\(_1\) and \(q <3/2\) (follows from (1.7)\(_1\)), we find, using also mean value theorem
In turn, using in order: the minimality of \(u_{\omega ,\varepsilon }\), (5.1) and (5.6), we have
so that, using the content of the last two displays we gain
Estimate (5.8) and (3.5)\(_1\) imply that for every \(\varepsilon \in (0,1)\) the sequence \(\{u_{\omega ,\varepsilon }\}_{\omega }\) is uniformly bounded in \(W^{1,q}(B_{r})\), therefore, up to not relabelled subsequences, we have
as \(\omega \rightarrow 0\). Letting \(\omega \rightarrow 0\) in (5.9) and using standard weak lower semicontinuity theorems, yields
for every fixed \(\varepsilon \in (0,1)\). Using (5.8) we conclude with
and again this holds for every \(\varepsilon \in (0,1)\). By (1.11)\(_1\) and (5.11) the sequence \(\{|Du_{\varepsilon }|g(Du_{\varepsilon })\}\) is uniformly bounded in \(L^{1}(B_{r})\). Recalling that the assumptions on \(g(\cdot )\) imply that \(g(t)\rightarrow \infty \) as \(t\rightarrow \infty \), by classical results of Dunford & Pettis and de la Vallée Poussin, there exists \(\hat{u}\in W^{1,1}(B_{r})\) such that \( u_{\varepsilon }\rightharpoonup \hat{u}\) weakly in \(W^{1,1}(B_{r})\) and \(\hat{u}-u \in W^{1,1}_0(B_{r})\). Letting \(\varepsilon \rightarrow 0\) in (5.11) weak lower semicontinuity (see [37, Theorem 4.3]) and (5.2) yield 
, while the opposite inequality follows by the minimality of u. We conclude with 
, so that, by strict convexity of the functional 
we find that \(u\equiv \hat{u}\) in \(B_{r}\) and we deduce that
5.3 Blow-Up
We fix \(\omega , \varepsilon \in (0,1]\) and \(u_{\omega ,\varepsilon }\in W^{1,q}(B_{r})\) as in (5.3). With \(B_{\varrho }(x_{0})\Subset B_{r}\) being a ball not necessarily concentric to \(B_{r}\), we take \(\mathfrak {M}\) such that
where \(\tilde{E}_{\omega ,\sigma _{\varepsilon }}(\cdot )\) is defined in (3.13). The above quantities are finite by (5.5). We rescale u and \(H(x,z)=F(x,z)+a(x)|z|^q\) on \(B_{\varrho }(x_{0})\) defining
with 
. Note that, obviously, 
is still an integrand of the type in (3.1) (see also (5.15) below) and therefore the content of Section 3 applies to 
as well. Since \(u_{\omega , \varepsilon }\) solves (5.3), recalling the notation fixed in (3.2), it follows that 
is a local minimizer on 
of the functional
where, recalling the notation in (2.7)\(_1\), with 
it is
From now on, keeping fixed the choice of \(\omega , \varepsilon \) made at the beginning, in order to simplify the notation we shall omit to specify dependence on such parameters, simply abbreviating
for 
. The minimality of \(u_{\varrho }\) implies the validity of the Euler-Lagrange equation
By (3.5) the integrand \(H_{\varrho }(\cdot )\) satisfies
for any 
and all \(z,\xi \in {\mathbb {R}}^{n}\), where \(c\equiv c (\texttt {data} )\) and, according to the definitions in (3.4) and the notation in (5.16), we are denoting
5.4 Minimal Integrands
Here we are going to play with auxiliary functionals whose integrands are of the type in (3.2)\(_2\), and therefore featuring no explicit dependence on x (these are usually called “frozen” integrands). The results of Section 4 can be therefore applied. Let us fix a number \(\beta _{0}\in (0,1)\), to be determined in a few lines, and a vector \(h\in {\mathbb {R}}^{n}{\setminus } \{0\}\) such that
We take 
and fix a ball centered at \(x_{\textrm{c}}\) with radius \(|h|^{\beta _{0}}\), denoted by \(B_h\equiv B_{|h|^{\beta _{0}}}(x_{\textrm{c}})\). By (5.20) we have 
. We set
According to the notation established in (2.7), (3.2) and (5.15)–(5.16), we define
Note that \(\partial _z H_{\varrho ,\text {i}}\in C^{1}({\mathbb {R}}^{n};{\mathbb {R}}^{n})\). As in (3.7), with
from (1.12) and (3.8) it follows that
hold for all \(z,z_{1},z_{2},\xi \in {\mathbb {R}}^{n}\), with \(c\equiv c(\texttt {data} )\). Consistently with (2.8), here we are denoting
Recalling (3.11) and (5.19), we define, for \(t\ge 0\) and 
that,
with related explicit expressions as in (3.12) and
By (3.15) we have
and (5.13) implies
Looking at (5.23), Direct Methods and strict convexity provide us with a unique minimizer \(v\in u_{\varrho }+W^{1,q}_{0}(8B_h)\) defined by
so that, (5.23)\(_1\) obviously implies
and minimality gives
Lemma 5.2
Let \(\delta _{1} \in (0,1)\) and let \(\mu _n(\cdot )\) be the (non-decreasing) function introduced in Proposition 4.1. If \(1\le \mu < \mu _n(\delta _{1}/2)\), then the inequalities
and
hold with \(c\equiv c(\texttt {data} ,\delta _{1})\) and \(\mathfrak {m}\) is defined in (5.21).
Proof
The integrand \(H_{\varrho ,\text {i}}(\cdot )\) is of the type considered in Proposition 4.1 (compare with (5.22) and (5.23)), and we apply this last result to v, with \(B_\tau \equiv 8B_h\). It follows that for every \(\delta _{1} >0\)
holds provided \(1\le \mu < \mu _n(\delta _{1}/2)\le \mu _n(\delta _{1}/q)\), where \(c\equiv c(\texttt {data} ,\delta _{1})\). Here, as in (3.2)\(_3\), it is \( {\mathbb {H}}_{\varrho ,\text {i}}(t):= tg(t)+{\tilde{a}}_{\varrho , \text {i}}(8B_h)[t^2+\omega ^2]^{q/2}+1 \) for \(t\ge 0\) and in fact in (5.33) we have used (3.3). From (5.33) we can derive (5.31) using (1.12) with \(\varepsilon =q-1>0\). Next, we apply (4.7), which gives
so that (5.32) follows using (5.31) in the above inequality and observing that \((1+\delta _1)\delta _1/2 \le \delta _1\). \(\square \)
5.5 A Comparison Estimate
This is in the following:
Lemma 5.3
Let 
be as in (5.14), \(v \in u_{\varrho }+W^{1,q}_{0}(8B_h)\) as in (5.28) and \(\delta _{2} \in (0,1/2)\). The inequality
holds with \(c\equiv c(\texttt {data} ,\delta _{2})\), where \({\tilde{\alpha }}=\min \{\alpha , \alpha _0\}\), 
is as in (5.24), and \(\mathfrak {M}\) is any number satisfying (5.13) and therefore (5.27).
Proof
From (5.21) and (5.27), and yet recalling (3.16), we deduce
where \(c\equiv c(\mu , q, \Vert a\Vert _{L^\infty })\) in (5.35)\(_1\). We have
where \(c\equiv c(\texttt {data} ,\delta _{2})\). We now estimate the four integrals stemming from the second-last line in the above display. As \(\delta _{2}< 1/2\), we have
and in the last line we have used (5.35)\(_2\). Similarly, this time using (5.35)\(_1\), we find
Next, note that \(\delta _{2} < 1/2\) and \(q<3/2\) (follows from (1.7)\(_1\)) implies \(2-q-\delta _{2}/2>0\) and therefore we can estimate
where we again used (5.35)\(_2\). By means of (5.35)\(_1\) we find that
Finally, observe that \(\delta _{2}< 2-q\) as \(q < 3/2\), so that
Merging the content of the last six displays yields (5.34) with the asserted dependence of the constant c. Notice that the dependence of the constants on \(\delta _{2}\) comes from (1.12) (that has been used with \(\varepsilon = \delta _{2}/2\)). Notice also that (5.34) holds whenever we are assuming (1.11) with \(\mu \in [1, 3/2)\). \(\square \)
5.6 A Fractional Caccioppoli Inequality via Nonlinear Atomic Type Decompositions
Lemma 5.4
(Fractional Caccoppoli inequality) Let \(u_{\omega ,\varepsilon }\in W^{1,q}(B_{r})\) be as in (5.3); fix numbers \(\delta _{1}\in (0,1)\), \(\delta _{2} \in (0,1/2)\) and a ball \(B_{\varrho }(x_{0})\Subset B_{r}\). The inequality
holds whenever
and provided that
where \(c\equiv c (\texttt {data} , \delta _{1}, \delta _{2}, \beta )\) and
The function \(\mu _{n}(\cdot )\) is the one defined in Lemma 5.2.
Proof
This will be obtained by a technique that works as an analogue of dyadic decomposition in Besov spaces, but using the functions \(v\equiv v(8B_h)\) in (5.28) as “atoms”; see Remark 3 below. We therefore divide the proof in two steps.
Step 1: Estimates on a single ball \(B_h\). Here we again use the notation and the results in Sections 5.4–5.5. In particular, here we again argue on a fixed ball \(B _h\). Our goal here is to prove estimate (5.43) below. We recall that basic properties of difference quotients yield
where \(\kappa \ge 0\) is any number (recall that \(|h|\le |h|^{\beta _0}\)). Using this in connection with (5.32) we have
with \(c\equiv c(\texttt {data} ,\delta _{1})\) and moreover (5.31) holds by (5.38)\(_1\); recall that \(\mathfrak {m}\equiv \mathfrak {m}(8B_h)\) is defined in (5.21). Use of (5.32) is legitimate here as (5.38)\(_1\) is assumed. Let us recall that here it is \(E_{\varrho ,\text {i}}(|z|)\equiv E_{\varrho ,\text {i}}(|z|;8B_h)\). Let us now record a couple of auxiliary estimates. The first is obtained as
for \(c\equiv c(n,q)\). For the second auxiliary inequality, we estimate
where \(c\equiv c (\texttt {data} , \delta _{1})\) and in the last line we have also used (5.35)\(_1\). Using (5.35)\(_2\), we get that
Using this last estimate with (5.34), we conclude with
where \(c\equiv c(\texttt {data} , \delta _{1}, \delta _{2})\), that is the second auxiliary estimate we were aiming at. Triangle inequality now yields
Using the standard property of translations
valid for every \(g \in L^2(4B_h)\), (for this note that \(h + B_h \subset 4B_h\) as \(|h|\le 1\)), and also the Lipschitz continuity of truncations, that is \( |(s-\kappa )_{+}-(t-\kappa )_{+}|\le |s-t|\) for every \( s, t\in {\mathbb {R}}\), we continue to estimate as follows:
Have \(c\equiv c(\texttt {data} ,\delta _{1})\). Note that we have used the elementary estimate
which follows from the very definition of \(E_{\varrho ,\text {i}}(\cdot )\) in (5.25)\(_2\). By then using (5.41) and (5.42) to estimate the last two integrals in the above display, respectively, and again (5.35)\(_2\), we come to
with \({\tilde{c}}\equiv {\tilde{c}}(\texttt {data} , \delta _{1}, \delta _{2})\), where \(\mathfrak s_1\), \(\mathfrak s_2\) and \(\mathfrak s_3\) are as in (5.39) and we have taken \(\beta _0\) such that
Step 2: Patching estimates (5.43) on different balls. In this second and final step we are now going to recover estimates for \(\tau _{h}(E_{\varrho }(\cdot , |Du_{\varrho }|)-\kappa )_{+}\) on 
by patching up estimates (5.43) via a dyadic covering argument. This goes as follows: we take a lattice of cubes \(\{Q_{\gamma }\}_{\gamma \le \mathfrak {n}}\) with sidelength equal to \(2|h|^{\beta _{0}}/\sqrt{n}\), centered at points 
, with sides parallel to the coordinate axes, and such that
This family of cubes corresponds to a family of balls in the sense that \(Q_{\gamma }\equiv Q_{\text {inn}}(B_{\gamma })\) and \(B_{\gamma }:= B_{|h|^{\beta _{0}}}(x_{\gamma })\), as defined above. By construction, and in particular by (5.20), it is 
for all \(\gamma \le \mathfrak {n}\) and \( \mathfrak {n}\approx |h|^{-n\beta _{0}} \), where the implied constant depends on n. Moreover, by (5.44), each of the dilated balls \(8B_{\gamma _{t}}\) intersects the similar ones \(8B_{\gamma _{s}}\) fewer than \(\mathfrak {c}_n\) times, that is a number depending only on n (uniform finite intersection property). This implies that
holds for every Borel measure \(\lambda (\cdot )\) defined on 
. We then write estimates (5.43) on balls \(B_h\equiv B_{\gamma }\) and sum up in order to obtain
Thanks to the inequality in the last display we apply Lemma 2.3. This yields 
for all \(\beta \in (0,\alpha _{*})\) with the related a priori bound
which holds for every \(\chi \equiv \chi (\beta )\) as in (5.37), where \(c\equiv c(\texttt {data} , \delta _{1}, \delta _{2}, \beta )\). Notice that here we have used, in order, first (2.10), as a consequence of (5.46), and then (2.9). Scaling back from 
to \(B_{\varrho }\) in (5.47) via (5.14), (5.19) and (5.25), squaring the resulting inequality, and restoring the original notation, we arrive at (5.36) and the proof is complete. \(\square \)
Remark 3
(Nonlinear atoms) Here we briefly expand on the possible analogy between the classic atomic decompositions in fractional spaces (see for instance [2, Section 4.6] and [67, Chapter 2]) and the construction made in the proof of Lemma 5.4. Atomic decompositions of a Besov function w usually go via decompositions of the space \({\mathbb {R}}^n\) in dyadic grids with mesh \(2^{-k}\), \(k\in {\mathbb {N}}\), corresponding to the annuli (in the frequency space) considered in Littllewood-Paley theory. On each cube \(Q_{k,\gamma }\) of the grid one considers an atom \(a_{k,\gamma }(\cdot )\), i.e., a smooth function with certain control on its derivatives, up to the maximal degree of regularity one is interested in describing for w. Specifically, one requires that
hold for sufficiently large multi-indices S. Summing up (over \(\gamma \)) such atoms multiplied by suitable modulating coefficients, and then yet over all possible grids \(k\in {\mathbb {N}}\), allows to give a precise description of the smoothness of the function w. Such “linear” decompositions, although very efficient, are of little use when dealing with nonlinear problems as those considered in this paper. The idea in Lemma 5.4 is then, given a grid of size \(|h|^{\beta _0} \approx 1/2^{k\beta _0}\), and therefore a certain “height” in the frequency space, to consider atoms v that are in a sense close to the original solution u in that they are themselves solutions to nonlinear problems (with frozen coefficients). In other words we attempt a decomposition of the type
where \(v_{\gamma }\) is defined as in (5.28), with \(B_{\gamma }\equiv B_{h}\) as in Step 2 from Lemma 5.4 (\( \mathbb {1}_{B_\gamma }\) is the indicator function of the ball \(B_\gamma \)). Notice in fact the analogy with the second information in (5.48), describing the maximal smoothness of a classical atom, with the Caccioppoli inequality (5.32), from which one infers its fractional version for \(u_{\varrho }\), that is (5.36).
5.7 \(C^{0,1}\)-Bounds via Nonlinear Potentials
Here we deliver
Proposition 5.5
Let \(u_{\omega ,\varepsilon }\in W^{1,q}(B_{r})\) be as in (5.3). The inequality
holds whenever \(B_{t}\Subset B_s\subseteq B_{r}\) are concentric balls such that \(r\le 1\), where \(c\equiv c(\texttt {data} )\) and \(\vartheta \equiv \vartheta (n,q,\alpha , \alpha _0)\). The function \(\tilde{E}_{\omega ,\sigma _{\varepsilon }}(\cdot )\) has been introduced in (3.13).
Proof
In what follows, we are going to use Lemma 5.4 with \(\delta _1\in (0,1)\) whose size will determined in a few lines as a function of \(n,q,\alpha , \alpha _0\), and with \(\delta _2\) by now fixed by
Note that \(\delta _2>0\) follows as a consequence of the assumed bound (1.17) and that the choice in (5.51) makes \(\delta _2\) depending only on \(n,q,\alpha , \alpha _0\). Without loss of generality we can assume that
otherwise (5.50) is obvious. We consider concentric balls \(B_{t}\Subset B_{\tau _{1}}\Subset B_{\tau _{2}}\Subset B_{s}\) and set \(r_{0}:=(\tau _{2}-\tau _{1})/8\). It follows that \(B_{2r_{0}}(x_{0})\Subset B_{\tau _{2}}\) holds whenever \(x_0\in B_{\tau _1}\). Notice that by (5.5) every point is a Lebesgue point for both \(|Du_{\omega ,\varepsilon }|\) and \(E_{\omega ,\sigma _{\varepsilon }}(\cdot ,|Du_{\omega ,\varepsilon }|)\). By Lemma 5.4, and (5.36) used on \(B_{\varrho }(x_0)\subset B_{r_0}(x_0)\), we can apply Lemma 2.5 on \(B_{r_{0}}(x_{0})\) verifying (2.14) with
As \(x_0 \in B_{\tau _1}\) has been chosen arbitrarily, (2.15), with the choices in (5.53), implies that
for \(c\equiv c(\texttt {data} , \delta _{1})\). Recalling the definitions in (3.12)–(3.13), and the choice of \(\mathfrak {M}\) in (5.53), after a few elementary manipulations in (5.54) we arrive at
with \(c\equiv c(\texttt {data} , \delta _{1} )\). We now take \(\delta _{1}\) to be such that
Recalling (5.51), the quantity \(\delta _{1}\) is now determined as a function of the parameters \(n,q,\alpha , \alpha _0\). This determines a first restriction on the size of \(\mu \) via (5.38) and the choice of \(\delta _{1}\). Notice that at this stage the number \(\mu _{*}:=\mu _n(\delta _{1}/2)\), determining an upper bound on \(\mu \), depends on \(n,q,\alpha , \alpha _0\); notice also that all the above computation remain valid provided \(1\le \mu < \mu _{*}\equiv \mu _{*}(n,q,\alpha , \alpha _0)\). Finally, by (5.56) and a standard continuity argument we can further restrict the value of \(\mu \) finding \(\mu _{\text {max}}\le \mu _{*}\) such that
Notice also that the functions \(\mu \mapsto \mathfrak {s}_{1}(\mu ,\cdot ), \mathfrak {s}_{2}(\mu ,\cdot ), \mathfrak {s}_{3}(\mu ,\cdot )\) are increasing. This finally fixes the value of \(\mu _{\text {max}}\) from the statement of Theorem 1.3, with the asserted dependence on the constants. We now want to the estimate the potential terms appearing in the right-hand side of (5.55) by means of Lemma 2.4. In this respect, notice that \( n>2\alpha \) allows to verify (2.12), while we can take \(\gamma =1\) in (2.13) because
Applying (2.13) gives
with \(c\equiv c(n,q,\alpha , \alpha _0)\). Using (5.58) in (5.55), and recalling (5.57), we can use Young’s inequality to finally get
with \(c\equiv c(\texttt {data} )\) and \(\vartheta \equiv \vartheta (n,q,\alpha , \alpha _0)\ge 1\). This allows to apply Lemma 2.2 that provides
from which (5.50) follows using (3.17) and a few elementary manipulations. \(\square \)
5.8 Proof of (1.14)
Keeping in mind the notation of Proposition 5.5, (5.8) and (5.50) gives that
so that, recalling (3.13), we find that
It follows that, up to not relabelled subsequences, the convergence in (5.10) can be upgraded to \(u_{\omega ,\varepsilon }\rightharpoonup ^{*} u_{\varepsilon }\) in \( W^{1,\infty }(B_{t})\) for every \(\varepsilon >0\). Letting \(\omega \rightarrow 0\) in (5.59) yields
which again holds for every \(\varepsilon >0\), with \(c\equiv c (\texttt {data} )\) and \(\vartheta \equiv \vartheta (n,q,\alpha , \alpha _0)\). Similarly, as after (5.59), by (5.60) the convergence in (5.12) can be upgraded to \(u_{\varepsilon }\rightharpoonup ^{*} u\) in \( W^{1,\infty }(B_{t})\), so that (1.14) (for \(\texttt {s} =0\)) follows letting \(\varepsilon \rightarrow 0\) in (5.60), taking \(t=r/2\), recalling that \(\mu <3/2\) and renaming \(2\vartheta \) into \(\vartheta \).
5.9 Local Gradient Hölder Continuity
Since the result is local, up passing to smaller open subsets we can assume with no loss of generality that 
is finite (see the comment after (1.15)).
We fix an open subset \(\varOmega _0 \Subset \varOmega \) and the radius \(r:=\min \{\,\textrm{dist}(\varOmega _0, \partial \varOmega )/8, 1\}\). We take \(B_{2r}\) centred in \( \varOmega _0\) and recall that the quantities \(\mathfrak {m}_{\omega ,\varepsilon }\equiv \mathfrak {m}_{\omega ,\varepsilon }(r;B_{2r})\) and \(\mathfrak {m}_{\varepsilon }\equiv \mathfrak {m}_{\varepsilon }(r;B_{2r})\) are defined in (5.59) and (5.60), respectively. With \(\tau \le r\) and \(B_{\tau }\subset B_{r}\) being a ball concentric to \(B_{2r}\), we define \(v\in u_{\omega ,\varepsilon }+W^{1,q}_{0}(B_{\tau })\) as
so that
holds with \({\bar{c}}\equiv {\bar{c}} (\texttt {data} )\). The derivation of (5.61) follows using first (5.59) (with \(B_{r}\) replaced by \(B_{2r}\) and \(t=r\)) and combining it with (4.5) and easy manipulations as in Proposition 4.1 and Lemma 5.2. By Proposition 5.6 in the subsequent step and (5.61), we find that
holds for every \(\varrho \le \tau /2\). Here, in the notation of Proposition 5.6 below, it is \(c_{\omega ,\varepsilon }:= c_*(2{\bar{c}} \mathfrak {m}_{\omega ,\varepsilon }^2)\) and \(\beta _{\omega ,\varepsilon }:= \beta _*(2{\bar{c}} \mathfrak {m}_{\omega ,\varepsilon }^2)\); these constants are non-decreasing and non-increasing functions of \(\mathfrak {m}_{\omega ,\varepsilon }\), respectively. Letting first \(\omega \rightarrow 0\) and then \(\varepsilon \rightarrow 0\), we have that
(see (5.60)). In particular,
holds with \(c\equiv c (\texttt {data} )\) and \(\vartheta \equiv \vartheta (n,q,\alpha , \alpha _0)\) so that \(\mathfrak {m}_0\) only depends on \(\texttt {data} , \,\textrm{dist}(\varOmega _0, \partial \varOmega )\) and 
. Again letting \(\omega \rightarrow 0\) and then \(\varepsilon \rightarrow 0\), and passing to not relabelled subsequences, we can assume that \(c_{\omega ,\varepsilon }\rightarrow : = c_{\varepsilon }\rightarrow =: {c} \le c_*(4{\bar{c}} \mathfrak {m}^2) < \infty \). Notice that by (5.63) we have \( {c} \le c_*(4{\bar{c}} \mathfrak {m}_0^2)\) and this last quantity depends only on \(\texttt {data} , \,\textrm{dist}(\varOmega _0, \partial \varOmega )\) and 
but it is otherwise independent of the chosen ball \(B_{2r}\). In the following, with some abuse of notation, we keep on denoting by \(c_{\omega ,\varepsilon }\ge 1\) a double-sequence of constants with the above property, typically being itself released via a non-decreasing function of \(c_*(2{\bar{c}} \mathfrak {m}_{\omega ,\varepsilon }^2)\); the exact value of the numbers \(c_{\omega ,\varepsilon }\) may vary from line to line. A similar reasoning can be done for the exponents \(\beta _{\omega ,\varepsilon }\), that is, we have \(\beta _{\omega ,\varepsilon }\rightarrow : = \beta _{\varepsilon }\rightarrow =: \beta \ge \beta _*(4{\bar{c}} \mathfrak {m}^2) \ge \beta _*(4{\bar{c}} \mathfrak {m}_0^2)=: \tilde{\beta }\in (0,1)\). Similarly, \(\tilde{\beta }\) depends only \(\texttt {data} , \,\textrm{dist}(\varOmega _0, \partial \varOmega )\) and 
and is independent of the ball \(B_{2r}\) considered. Proceeding as for (5.34), and also using (5.61) repeatedly, it follows that
where \({\tilde{\alpha }}=\min \{\alpha , \alpha _0\}\). Using (2.4) with \(p=2-\mu \) and then (5.61) yields
By using (5.64) again in the above display, we conclude with
We can now complete the proof with more standard arguments (see for instance [35, 55, 56]), that we recall for completeness. Combining (5.62) and (5.65) yields
for every \(\varrho \le \tau /2\), and taking \(\varrho = (\tau /2)^{1+{\tilde{\alpha }}/(n+2\beta _{\omega ,\varepsilon })}\) finally yields
for every \(\varrho \le (r/2)^{1+{\tilde{\alpha }}/(n+2\beta _{\omega ,\varepsilon })}\). In the above display we first let \(\omega \rightarrow 0\) and then \(\varepsilon \rightarrow 0 \), and finally conclude with
where, by the discussion made after (5.63), both \(c\ge 1\) and \(\tilde{\beta }\in (0,1)\) depend only on \(\texttt {data} \), \(\,\textrm{dist}(\varOmega _0, \partial \varOmega )\) and 
, but are otherwise independent of the ball considered \(B_{2r}\). Since \(\varOmega _0\) is arbitrary, (5.66) and a standard covering argument and Campanato-Meyers integral characterization of Hölder continuity yield that for every open subset \(\varOmega _0 \Subset \varOmega \) it holds that \( [Du]_{0,\beta ; \varOmega _0}<\infty \), where \(\beta := \tilde{\alpha }\tilde{\beta }/(n+2\tilde{\beta }+\tilde{\alpha }) \). The proof of Theorem 1.3 in the case \(\texttt {s} =0\) is therefore complete up to (5.62), whose proof will be given in the subsequent section. We now briefly comment on how to obtain the local gradient Hölder continuity in the nonsingular case \(\texttt {s} >0\), which is simpler since it requires no approximation via the additional parameter \(\omega \). For the proof of (1.14) it is sufficient to use functionals 
as in (5.4), and all the subsequent estimates remain independent of \(\texttt {s} \) and \(\varepsilon \); the approximation and the convergence then only take place with respect to the parameter \(\varepsilon \) and leads to (1.14). The same reasoning applies to the proof in this section by taking \(\omega \equiv \texttt {s} \). It remains to deal with the last issue, that is the local Hölder continuity of Du with explicit exponent when \(\texttt {s} >0\). This will be done in Section 5.11.
5.10 A Technical Decay Estimate
Here we prove (5.62); this relies on certain hidden facts from regularity theory of singular parabolic equations that we take as a starting point to treat the nonuniformly elliptic case considered here. Relevant related methods are also in [26]. Consistently with the notation established in Sections 3 and 4, with \({\bar{a}} > 0\) being a fixed number, in the following we denote that
for every \(z\in {\mathbb {R}}^n\), where \(\omega \in (0,1]\) and \(x_{\textrm{c}}\in \varOmega \). Here we permanently assume (1.11)–(1.13) and therefore \(H_{\text {i}}(\cdot )\) is an integrand of the type in (3.2)\(_2\) with \(a_{\sigma , \text {i}}(B)\) replaced by \({\bar{a}}\). Indeed, conditions (3.8) apply here when accordingly recasted (see also (5.78) below). Moreover, let us define
Then (1.13) and Lemma 2.1 imply that
holds for every \(z \in {\mathbb {R}}^n\), where \(c\equiv c (q,L, c_g(1))\).
Proposition 5.6
Under assumptions (1.11)–(1.13), with \(B\subset {\mathbb {R}}^n\) being a ball, let \(v\in W^{1,\infty }(B)\) such that
There exist two functions of \(\mathfrak {m}\), \(c_*:[1, \infty )\rightarrow [1, \infty )\) and \(\beta _*:[1, \infty ) \rightarrow (0,1)\), such that
holds for every \(\tau \in (0,1)\). The functions \(c_*(\cdot )\) and \(\beta _*(\cdot )\) are non-decreasing and non-increasing, respectively, and also depend on \(n,q, \mu ,\nu , L, g(\cdot )\). They are independent of \({\bar{a}}\) and \(\omega \in (0,1]\).
The proof of (5.71) is achieved via two preliminary lemmas. In the rest of section, we denote \(\Vert Dv\Vert := \max _{1\le s\le n}|D_s v|\).
Lemma 5.7
With \(B\subset {\mathbb {R}}^n\) being a ball, assume that \(v\in W^{1,\infty }(B)\) satisfies (5.70)\(_1\) and
There exists \(\sigma \equiv \sigma (n,\mu , q, \nu , L, c_{g}(1), \mathfrak {m}, g(\mathfrak {m}))\in (0,1)\), which is independent of \({\bar{a}}\) and is a non-increasing function of \(\mathfrak {m}\), such that if either
holds for some \(s \in \{1, \ldots , n\}\), then
In this case
where \(c_{1}\ge 1\) and \(\beta _{1} \in (0,1)\) both depend on \(n,\mu , q, \nu , L,\mathfrak {m}, g(\mathfrak {m})\), but are otherwise independent of \({\bar{a}}\). The constants \(c_{1}\) and \(\beta _1\) are non-decreasing and non-increasing functions of \(\mathfrak {m}\), respectively. On the other hand there exists \(\tilde{\eta }\equiv \tilde{\eta }(n,\mu , q,\nu , L, \mathfrak {m}, g(\mathfrak {m})) \in (1/2,1)\) such that, if \(\texttt {ex} (B, \lambda , s)\) fails for every \(s \in \{1, \ldots , n\}\), then
The constant \(\tilde{\eta }\) is a non-decreasing function of \(\mathfrak {m}\).
Proof
For the proof of the first assertion we use and extend some of the arguments in [53, Proposition 3.7]; see also [26] for the case of the classical p-Laplacian operator. We can of course confine ourselves to the case the first inequality in (5.73) occurs (for a \(\sigma \) to be determined in the course of the proof), the other being similar. By scaling, as in (5.14), we can assume that 
. Since conditions (3.10) are satisfied by \(H_{\text {i}}(\cdot )\), it is standard to prove that 
. We can then differentiate the equation \(\,\textrm{div}\, \partial _z H_{\text {i}} (Dv)=0\) in the \(x_s\) direction, thereby obtaining \(\,\textrm{div}\, (\partial _{zz} H_{\text {i}} (Dv)DD_sv)=0\), that is an analogue of (4.8). Recalling that 
(as in (4.4) and by (5.72)), we then use as test function \((D_s v-\kappa )_{-}\phi ^2\), where 
is non-negative and \(0 \le \kappa \le \lambda \); this is still possible by (3.10)). Integrating by parts the resulting equation we find
We define the functions
for \(t\ge 0\), so that
holds for \(c\equiv c (q)\). By also using (3.8), (5.69) and (5.72), and the monotonicity features of \( \lambda _{{\bar{a}}}(\cdot )\) and \(G_0(\cdot )\) in (5.77), we easily find
By then using Young’s inequality and (5.79), and yet recalling that \((D_s v-\kappa )_{-}\le 2\lambda \), we find
For integers \(m\ge 0\) we choose levels \(\{\kappa _m=\lambda (1+1/2^m)/4\}\), radii \(\{\varrho _m=1/2+1/2^{m+1}\}\) and cut-off functions 
, with \(\phi _m\equiv 1\) on 
, \(\Vert D\phi _m\Vert _{L^\infty }^2+\Vert D^2\phi _m\Vert _{L^\infty } \lesssim 4^m\). With
estimate (5.80) becomes
We then find, via Sobolev embedding that,
Observing that \(\kappa _{m}-\kappa _{m+1}\approx 2^{-m}\lambda \) and that (5.72) implies \(\lambda ^2\approx \lambda ^2+\omega ^2\), the above display gives
with \(c\equiv c(n,\mu , q, \nu , L,c_{g}(1))\) which is independent of \({\bar{a}}\). Inequality (5.82) allows to perform a standard geometric iteration (see for instance [37, Lemma 7.1]) leading to \( D_s v\ge \lambda /4\) once the first inequality in (5.73) is verified with \(\sigma \equiv \sigma (n,\mu , q, \nu , L, c_{g}(1),\mathfrak {m}, g(\mathfrak {m}))\) small enough (that corresponds to require that \(|A_0|\) is small, see (5.81)). This allows to prove (5.75) by considering that
this time for every \(t \in \{1, \ldots , n\}\). Recalling (3.8)–(3.9) it follows that \(\texttt {a}(\cdot )\) satisfies
a.e. in 
and for every \(\xi \in {\mathbb {R}}^n\), where \(c \equiv c (n,\mu , q, \nu , L)\). Therefore a standard application of De Giorgi-Nash-Moser theory to \(D_tv\) and (5.72) imply the validity of (5.75) (note that for this we obtain (5.75) for \(\tau \le 1/8\), and then the case \(1/8< \tau \le 1\) follows trivially by (5.72)). We now turn to (5.76); the proof is a variant of the one in [53, Proposition 3.11] apart from the parabolic case considered there. The only remark we need to do is that the analogue of equation in [53, (3.53)] is here given by \(\,\textrm{div}\, (\texttt {b}(x)Dz)=0\), where \(\texttt {b}(x):= [\lambda _{{\bar{a}}}(\lambda ) ]^{-1} \partial _{zz}H_{\text {i}}(Dv(x))=\texttt {a}(x)\) if x belongs to the support of \({\tilde{v}}:= (D_sv-\lambda /2)_{+}\), and \(\texttt {b}(x)\equiv \mathbb {I}_d\) (the identity matrix) otherwise. Then \({\tilde{v}}\) turns out to be a weak subsolution. Moreover, as \(|Dv| \ge \lambda /2\) on the support of \({\tilde{v}}\), it follows that \(\texttt {b}(x)\) also satisfies (5.84). We can now use the arguments in [53, pp. 784–786] to conclude with (5.76). Finally, as for the monotone dependence on \(\mathfrak {m}\) of the constants \(c_{1}, \beta _{1}, \tilde{\eta }, \sigma \), this is classical, and it is a consequence of the fact that all the estimates above feature constants, usually denoted by c, that are non-decreasing functions of \(\mathfrak {m}\) and \(g(\mathfrak {m})\); in turn, \(g(\cdot )\) is a non-decreasing function too. \(\square \)
Lemma 5.8
Let v be as in Lemma 5.7, assume (5.72) and that \(m \omega \ge \lambda \) holds for some integer \(m\ge 1\). Then
where \(c_{2}\ge 1\) and \(\beta _{2} \in (0,1)\) both depend on \(n,\mu , q, \nu , L,\mathfrak {m},g(\mathfrak {m}), m\), but are otherwise independent of \({\bar{a}}\). The constants \(c_{2}\) and \(\beta _2\) are non-decreasing and non-increasing functions on \(\mathfrak {m}\) and m, respectively.
Proof
As for Lemma 5.7 we reduce to the case 
and consider (5.83). The first inequality in (5.84) follows as \(\Vert Dv\Vert \le \lambda \) a.e.; using \(m \omega \ge \lambda \) and (3.8)\(_3\) we instead have
where \(c \equiv c (n,\mu , q, \nu , L)\). Now (5.85) follows from standard De Giorgi-Nash-Moser theory. \(\square \)
Proof of Proposition 5.6
We take \(\sigma , \tilde{\eta }\equiv \sigma , \tilde{\eta } (n,\mu , q, \nu , L, \mathfrak {m}, g(\mathfrak {m}))\in (0,1)\) from Lemma 5.7 and determine an integer \(m\equiv m (n,\mu , q, \nu , L, \mathfrak {m}, g(\mathfrak {m})) \ge 1\) such that \(\eta := \tilde{\eta }+ 1/m<1\); notice that m can be determined as a non-decreasing function of \(\mathfrak {m}\). All in all, once \(\mathfrak {m}\) is fixed so is m and therefore \(\eta \). We set \(\lambda _0:= \sup _{B}\, \Vert Dv\Vert +\omega \), \(\lambda _k:= \eta ^{k} \lambda _0\), \(B_k:=B/2^k\) for every \(k \in {\mathbb {N}}\), and define \({\bar{k}} \in {\mathbb {N}}\) as the smallest non-negative integer for which \(m\omega > \lambda _{{\bar{k}}}\). This implies that
Note that (5.86) is empty when \({\bar{k}} =0\). We further define that
and \({\tilde{k}}\) as \({\tilde{k}}:= \min A\) if \(A\not =\emptyset \), and \({\tilde{k}}={\bar{k}}\) if \(A=\emptyset \). Using (5.76) and (5.86) iteratively, we find
On the other hand, by the very definition of \({\tilde{k}}\) and Lemmas 5.7–5.8 (the latter is only needed when \({\tilde{k}}={\bar{k}}\) and therefore, when \(A=\emptyset \)), we find that
At this stage (5.71) follows via a standard interpolation argument combining (5.87)–(5.88). \(\square \)
Remark 4
As a consequence of the local gradient boundedness obtained in Section 5.8, the arguments developed for Proposition 5.6 do not require any upper bound on \(\mu \). Moreover, the specific structure in (5.67) is not indispensable and the methods used here can be used as a starting point to treat general nonautonomous functionals with (p, q)-growth of the type considered in [60].
5.11 Improved Hölder Exponent when \(\texttt {s} >0\)
The \( C^{1, \tilde{\alpha }/2}_{\textrm{loc}}(\varOmega )\)-regularity proof follows almost verbatim the one in [24, Section 10] since now we already know that Du is locally Hölder continuous in \(\varOmega \) with exponent \(\beta \). We confine ourselves to give a few remarks on the main modifications and to facilitate the adaptation we use the same notation introduced in [24]. We set \(M:=\Vert Du\Vert _{L^\infty (\varOmega )}+[Du]_{0, \beta ;\varOmega }+1\) (we can assume this is finite since our result is local). This time we take \(A_r(z):= \partial _z H_{r^{{\tilde{\alpha }}},\text {i}}(z) = \partial _z F(x_{\textrm{c}}, z) + qa_{r^{{\tilde{\alpha }}}, \text {i}}(B_{r})[\ell _{\texttt {s} }(z)]^{q-2}z \) and recall that \(|\partial _z H_{r^{{\tilde{\alpha }}},\text {i}}(z)-\partial _z H(x, z)|\le cr^{{\tilde{\alpha }}}[\ell _{1}(z)]^{q-1}\) holds whenever \(x\in B_{r}\). With \(v \in u+W^{1,q}_0(B_{r})\) to solve \(\,\textrm{div}\, A_r(Dv)=0\) in \(B_{r}\), as in (5.61) we gain \( \Vert Dv \Vert _{L^{\infty }(B_{r/2})}\le c M^{2} \) with \(c\equiv c (\texttt {data} )\). With these informations estimate \( \Vert Du-Dv \Vert _{L^2(B_{r/2})}^2 \le cr^{n+{\tilde{\alpha }}}\) follows as in (5.65) (in turn as in (5.34)), with \(c\equiv c (\texttt {data} ,M)\). This is the analogue of [24, (10.3)], but for the fact that here we find \({\tilde{\alpha }}\) rather than \(2{\tilde{\alpha }}\) and the integral is supported in \(B_{r/2}\) instead of \(B_{r}\). Next, by defining the matrix \([{\mathbb {A}}(x)]_{ij}:= \partial _{z_j}A_{r}^i(Dv(x))\), to use [24] we need to prove that there exists \(\lambda \equiv \lambda (\texttt {data} , \texttt {s} , M)>0\), independent of r, such that \(\lambda \mathbb {I}_{\textrm{d}} \le {\mathbb {A}}(x)\le (1/\lambda )\mathbb {I}_{\textrm{d}}\) holds for a.e. \(x \in B_{r/2}\). As a consequence of (1.11)\(_2\) and of \(|Dv|\lesssim M^2\) in \(B_{r/2}\), we have
for a.e. \(x\in B_{r/2}\) and \(z, \xi \in {\mathbb {R}}^n\), where \(c\equiv (\texttt {data} , M)\). This is the crucial point where we use After this, the proof follows exactly as [24, (10.6)] and the proof of the entire Theorem 1.3 is finally complete.
6 Theorem 1.1, Corollary 1.2 and Model Examples (1.10)
For every integer \(k\ge 0\), we consider the integrand
with \(\mathfrak {c}(\cdot )\) as in (1.7)\(_2\) and \(L_{k+1}(\cdot )\) as in (1.10)\(_2\). Direct computations show that \(F(\cdot )\) satisfies (1.11) with \(g(t)\equiv L_{k+1}(t) +1\), with \(\mu =1\) when \(k=0\), and with any choice of \(\mu >1\) when \(k>0\); moreover, it also satisfies (1.13). The constants \(\nu \) and L depend on \(\mu , k\) and \(\varLambda \). Theorem 1.3 therefore applies to local minimizers of
On the other hand the functional in the above display and
share the same local minimizers, and therefore the regularity results stated in Theorem 1.3 hold for local minimizers of the last functional too. In this way the model case in (1.10) is covered. Theorem 1.1 and Corollary 1.2 then follow as special cases taking \(k=0\) and \(\texttt {s} =0\) and \(\texttt {s} =1\), respectively. Note that, in the spirit of Corollary 1.2, local minimizers of the functional
are locally \(C^{1,\alpha _0/2}\)-regular in \(\varOmega \), for every \(k\ge 0\), provided (1.7)\(_2\) is assumed.
7 Vectorial Cases and Further Generalizations
When functionals as in (1.1) and (1.9) are considered in the vectorial case, i.e. minima are vector valued \(u :\varOmega \rightarrow {\mathbb {R}}^N\) and \(N>1\), we can still obtain Lipschitz continuity results. In this situation it is unavoidable to impose a so-called Uhlenbeck structure [69], that is,
where \({\tilde{F}}:\varOmega \times [0, \infty ) \rightarrow [0, \infty )\) is a continuous function such that \(t \mapsto {\tilde{F}}(x, t)\) is \(C^2\)-regular for every choice of \(x\in \varOmega \). This is obviously satisfied by the models in (1.1) and (1.10).
Theorem 7.1
Let \(u\in W^{1,1}_{\textrm{loc}}(\varOmega ;{\mathbb {R}}^N)\) be a local minimizer of the functional in (1.9) under assumptions (1.7)\(_1\), (1.11)–(1.12) and (7.1). There exists \(\mu _{\text {max}}\in (1,2)\), depending only \(n,q, \alpha , \alpha _0\), such that, if \(1\le \mu < \mu _{\text {max}}\), then (1.14) holds as in Theorem 1.3.
The proof of Theorem 7.1 is essentially the same of the one given for Theorem 1.3, once the content of Proposition 4.1 is available. This is in fact the only point where (7.1) enters the game. Inspecting the proof of Proposition 4.1 shows that this works in the vectorial case provided (4.9) holds. In turn, this inequality follows along the lines of the estimates in [8, Lemma 5.6]. Notice that, without an additional structure assumption as in (7.1), Theorem 7.1 cannot hold and counterexamples to Lipschitz regularity emerge already when considering uniformly elliptic systems [66]. We also remark that we expect gradient Hölder continuity in the vectorial case as well; the proof must be different from the one given in Section 5.10, and based on linearization methods as those originally introduced in [69] and developed in large parts of the subsequent literature. Another direct generalization, this time in the scalar case, occurs for functionals of the type
where \(F(\cdot )\) is as Theorem 1.3 and \(G:{\mathbb {R}}^n\rightarrow [0, \infty )\) belongs to \(C^1({\mathbb {R}}^n)\cap C^2({\mathbb {R}}^n{\setminus }\{0\})\) and satisfies
for every \(z, \xi \in {\mathbb {R}}^n\), \(|z|\not = 0 \). Theorem 1.3 continuous to hold in this last case, with essentially the same proof. The only difference worth pointing out is the new shape of the integrands \(H(x, z):=F(x, z)+a(x)G(z)\) in (3.1) and \(H_{\omega ,\sigma }(\cdot )\) (and the related minimal one \(H_{\omega ,\sigma ,\text {i}}(\cdot )\)) in (3.2), accordingly defined as
Here \(\{\phi _{\varepsilon }\}\subset C^{\infty }({\mathbb {R}}^n)\), denotes a family of radially symmetric mollifiers, defined as \(\phi _{\varepsilon }(x):=\phi (x/\varepsilon )/\varepsilon ^n\), where 
, \( \Vert \phi \Vert _{L^{1}({\mathbb {R}}^n)}=1\), 
. The definition of \({\mathbb {H}}_{\omega ,\sigma ,\text {i}}(\cdot )\), instead, remains the same. Following for instance [22, Section 4.5] and [24, Section 6.7] it is possible to show that the newly defined integrands \(H_{\omega ,\sigma }(\cdot )\) still have the properties described in Section 3. In particular (3.5) and (3.6) still hold for a suitable constant c with the same dependence upon the various parameters described there. Finally, when considering again the vectorial case for functionals as in (7.2), we can still obtain local Lipschitz regularity of minima provided (7.1) holds together with \(G(z)\equiv {\tilde{G}}(|z|)\) that and \({\tilde{G}}(\cdot )\) is \(C^2\)-regular outside the origin (see for instance [8, 23] for precise assumptions).
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Acknowledgements
The first author is supported by INdAM-GNAMPA via the project “Problemi non locali: teoria cinetica e non uniforme ellitticità“, and by the University of Parma via the project “Local vs nonlocal: mixed type operators and nonuniform ellipticity”. Both of the authors are grateful to the referees for the careful reading of the original version of the manuscript and for the many suggestions and comments that eventually led to better presentation.
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De Filippis, C., Mingione, G. Regularity for Double Phase Problems at Nearly Linear Growth. Arch Rational Mech Anal 247, 85 (2023). https://doi.org/10.1007/s00205-023-01907-3
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DOI: https://doi.org/10.1007/s00205-023-01907-3