Regularity for Double Phase Problems at
Nearly Linear Growth

Theorem 1.1 and Corollary 1 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 $\textnormal{{s}}\in[0,1]$. 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 $F\colon\Omega\times\mathbb{R}^{n}\to[0,\infty)$ such that $z\mapsto F(x,z)\in C^{2}(\mathbb{R}^{n})$ for every $x\in\Omega$ and satisfying

for every choice of $x,y\in\Omega$, $z,\xi\in\mathbb{R}^{n}$, with $\alpha_{0}\in(0,1)$, $1\leq\mu<3/2$ and $0<\nu\leq 1\leq L$ being fixed constants. Here $g\colon[0,\infty)\to[1,\infty)$ 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

The notion of local minimizer we are using in this paper is classical and prescribes that a function $u\in W^{1,1}_{{\rm loc}}(\Omega)$ is a local minimizer of $\mathcal{N}(\cdot)$ if, for every ball $B\Subset\Omega$, it happens that

This implies, in particular, that $\mathcal{N}(u,\Omega_{0})$ is finite whenever $\Omega_{0}\Subset\Omega$ is an open subset. In Theorem 1.2, we are using the shorthand notation

that 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 $c\equiv c(\textnormal{{data}})$ a constant actually depending only on a subset of the parameters listed in (1.16).

Theorem 1.2 has a technical nature, especially as far as the size of $\mu_{\textnormal{max}}$ 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 $w\mapsto\int F(Dw)\,dx$, satisfying (1.11)_1,2 with restrictions as $1\leq\mu<1+\texttt{o}(n)$, $\texttt{o}(n)\to 0$ when $n\to\infty$. These bounds are in fact of the type we also need to impose when selecting $\mu_{\textnormal{max}}$ (this can be checked by tracking the constant dependence in the proofs). Starting by FM conditions (1.11)₂ have been used in several papers devoted to functionals with linear or nearly linear growth; see for instance bildhauer ; BF ; gmek2 and related references.

The techniques needed for Theorem 1.2 are entirely different from those used in the case of classical superlinear double phase functionals in (1.3) BCM ; CM and for pointwise uniformly elliptic problems with nonstandard growth conditions HO1 ; HO2 . This is essentially due to the failure of uniform ellipticity (1.5) for functionals as $\mathcal{L}(\cdot)$ and $\mathcal{N}(\cdot)$. 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 bildhauer ; BF2 ; FrS ; FM 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 piovra . 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 (10). 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 (10). The term renormalized accounts for the fact that inequalities (10) are homogeneous with respect to $E(\cdot,Du)$, despite the functional $\mathcal{N}(\cdot)$ is not. This costs the price of getting multiplicative constants depending on $\|Du\|_{L^{\infty}}$. Such constants must be carefully kept under control all over the proof and reabsorbed at the very end. Notice that inequalities (10) are obtained via a dyadic/atomic decomposition technique, finding its roots in KM , that resembles the one used for Besov spaces AH ; triebel in the setting of Littlewood-Paley theory; see Section 5.6 and Remark 3. Iterating inequalities (10) via a potential theoretic version of De Giorgi’s iteration (Lemma 5) then leads to establish local Lipschitz bounds in terms of the nonlinear potentials defined in (2.11); see Proposition 2. A crucial point is that the fractional nature of the inequalities (10) 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 $\mathcal{N}(\cdot)$ 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 KuM . 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.

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, $Q_{\textnormal{inn}}\equiv Q_{\textnormal{inn}}(B)$ 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 $B$, $Q_{\textnormal{inn}}(B)\subset B$. The sidelength of $Q_{\textnormal{inn}}(B)$ equals $2r/\sqrt{n}$. The vector valued version of function spaces like $L^{p}(\Omega),W^{1,p}(\Omega)$, i.e., when the maps considered take values in $\mathbb{R}^{k}$, $k\in\mathbb{N}$, will be denoted by $L^{p}(\Omega;\mathbb{R}^{k}),W^{1,p}(\Omega;\mathbb{R}^{k})$; when no ambiguity will arise we shall still denote $L^{p}(\Omega)\equiv L^{p}(\Omega;\mathbb{R}^{k})$ and so forth. For the rest of the paper we shall keep the notation

for $z\in\mathbb{R}^{k}$, $k\geq 1$ and $\omega\in[0,1]$. In particular, when $k=1$, we have a function defined on $\mathbb{R}$. With $\mathcal{U}\subset\mathbb{R}^{n}$ being a measurable subset with bounded positive measure $0<|\mathcal{U}|<\infty$, and with $f\colon\mathcal{U}\to\mathbb{R}^{k}$, $k\geq 1$, being an integrable map, we denote

Moreover, given a (scalar) function $f$ and a number $\kappa\in\mathbb{R}$, we denote

Finally, whenever $f\colon\mathcal{U}\to\mathbb{R}^{k}$ and $\mathcal{U}$ is any set, we define

As usual, with $\beta\in(0,1]$ we denote

We shall use the vector fields $V_{\omega,p}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$, defined by

whenever $z\in\mathbb{R}^{n}$ (compare (ha, , (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 (ha, , Lemma 2.1). We shall also use the following, trivial

that holds for every $\gamma\geq 0$ 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 ha , (giamod, , Section 2). With $B\subset\Omega$ being a ball and $\sigma\in(0,1)$, we let

and define

where $z_{1},z_{2}\in\mathbb{R}^{n}$, $x\in\Omega$. A consequence of assumptions (1.11) is the following:

Finally, a classical iteration lemma (giu, , Lemma 6.1).

The finite difference operator $\tau_{h}\colon L^{1}(\Omega;\mathbb{R}^{k})\to L^{1}(\Omega_{|h|};\mathbb{R}^{k})$ is defined as $\tau_{h}w(x):=w(x+h)-w(x)$ for $x\in\Omega_{\lvert h\rvert}$, for every $\tau\in L^{1}(\Omega)$ and where $\Omega_{|h|}:=\{x\in\Omega\,:\,\,{\rm dist}(x,\partial\Omega)>|h|\}$. With $\beta\in(0,1)$, $s\in[1,\infty)$, $k\in\mathbb{N}$, $n\geq 2$, and with $\Omega\subset\mathbb{R}^{n}$ being an open subset, the space $W^{\beta,s}(\Omega;\mathbb{R}^{k})$ is defined as the set of maps $w\colon\Omega\to\mathbb{R}^{k}$ such that

When considering regular domains, as for instance the ball $\mathcal{B}_{1/2}$ (this is the only case we are going to consider here in this respect), the embedding inequality

holds provided $s\geq 1,\beta\in(0,1)$, $k\in\mathbb{N}$ and $s\beta<n$. For this and basic results concerning Fractional Sobolev spaces we refer to guide 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 dm .

We shall use a family of nonlinear potentials that, in their essence, goes back to the fundamental work of Havin & Mazya HM . We have recently used such potentials in the setting of nonuniformly elliptic problems in BM ; camel ; camel2 ; ciccio ; piovra and in fact we shall in particular use (piovra, , Section 4) as a main source of tools in this respect. With $t,\sigma>0$, $m,\theta\geq 0$ being fixed parameters, and $f\in L^{1}(B_{r}(x_{0}))$ being such that $\lvert f\rvert^{m}\in L^{1}(B_{r}(x_{0}))$, where $B_{r}(x_{0})\subset\mathbb{R}^{n}$ is a ball, the nonlinear Havin-Mazya-Wolff type potential ${\bf P}_{t,\sigma}^{m,\theta}(f;\cdot)$ is defined by

Suitable integrability properties of $f$ guarantee that ${\bf P}_{t,\sigma}^{m,\theta}(f;\cdot)$ is bounded, as stated in

Note that in (2.13) we are not requiring that $\gamma\geq 1$ and any value $\gamma>0$ is allowed. Next lemma is a version of (piovra, , Lemma 2.3) and takes information from various results from Nonlinear Potential Theory starting by the seminal paper kilp . 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 $M_{0},M_{i}$.