Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

We are interested in the regularity of local minimizers of energy integrals of the Calculus of Variations. Precisely, let \begin{document}$Ω $\end{document} be an open subset of \begin{document}$\mathbb{R}^{n}$\end{document} . Let \begin{document}$f≤\left( {x, \xi } \right) $\end{document} be a real function defined in \begin{document}$Ω × \mathbb{R}^{n}$\end{document} satisfying the growth condition \begin{document}$|{f_{\xi x}}\left( {x, \xi } \right)| \le h\left( x \right)|\xi {{\rm{|}}^{p - 1}}$\end{document} , for \begin{document}$x∈ Ω $\end{document} and \begin{document}$\xi ∈ \mathbb{R}^{n}$\end{document} with \begin{document}$|\xi {\rm{|}} \ge {M_0}$\end{document} for some \begin{document}$M_{0}≥ 0$\end{document} , with \begin{document}$h \in L_{{\rm{loc}}}^r\left( \Omega \right) $\end{document} for some \begin{document}$r>n$\end{document} . This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to \begin{document}$f\left( {x, \xi } \right) $\end{document} the so-called natural \begin{document}$p-$\end{document} growth conditions on the second derivatives \begin{document}${f_{\xi \xi }}\left( {x, \xi } \right)$\end{document} ; i.e., \begin{document}$\left( {p - 2} \right) - $\end{document} growth for \begin{document}$|{f_{\xi \xi }}\left( {x, \xi } \right)| $\end{document} from above and \begin{document}$\left( {p - 2} \right) - $\end{document} growth from below for the quadratic form \begin{document}$({f_{\xi \xi }}\left( {x, \xi } \right)\lambda , \lambda {\rm{ }})$\end{document} ; for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer \begin{document}$u \in W_{{\rm{loc}}}^{1, p}\left( \Omega \right) $\end{document} of the energy integral \begin{document}$\int_\Omega {f(x, Du\left( x \right)){\mkern 1mu} dx} $\end{document} .


1.
Introduction. In recent years many authors considered differential problems, in the context of the calculus of variations and of partial differential equations of elliptic and parabolic type, under general p, q−growth conditions and, as a special relevant case, related to p (x) −growth; i.e., variable exponents. Among them Vicenţiu D. Rȃdulescu who studied, in this framework of general growth, multiplicity of solutions for some nonlinear problems, qualitative analysis, anisotropic elliptic equations, eigenvalue problems and several other related questions; see for instance [20], [21], [5], [23], [2]. We like to explicitly dedicate this manuscript to Vicenţiu D. Rȃdulescu, with esteem and sympathy.
In order to introduce the problem we first consider the classical Dirichlet energy integral and positive in Ω. In the nonlinear case too, more generally, a minimizer of an energy integral of the type Ω f (x, Du (x)) dx (2) comes out to be locally Lipschitz continuous in Ω if the integrand f : Ω × R n → R satisfies the ellipticity and the growth conditions for some positive constants M 1 , M 2 , for an exponent p ∈ (1, +∞) and for every x ∈ Ω and ξ ∈ R n . The assumption usually considered in the mathematical literature for Lipschitz continuity of solutions is the Lipschitz continuity of f (x, ξ) with respect to x; more precisely, similarly to the Dirichlet integral in (1), the condition often assumed on the x−dependence is A classical reference is, for instance, the book by Ladyzhenskaya-Uraltseva [19]. The x−dependence cannot be only considered as a perturbation, but it is a relevant difference with the case f = f (ξ) from several points of view. For instance recently several authors studied the x−dependence under Hölder continuity assumptions as well as under Sobolev summability assumptions, in the general context of p, q−growth conditions; see [7], [8], [13], [14]; see also [11], [3], [4], [15]. In this paper we show that we can obtain the local Lipschitz continuity in Ω of the local minimizers by assuming a mild condition on the x−dependence, weaker than (4). Precisely, instead of (4), we assume that where h ∈ L r loc (Ω) for some exponent r > n. A precise statement (with assumptions only for |ξ| → +∞) is described in the next section. The following regularity theorem holds. Theorem 1. Let p > 1 and u ∈ W 1,p loc (Ω) be a local minimizer of the energy integral (2) under the growth assumptions (3), (5). Then u is locally Lipschitz continuous in Ω.
An explicit bound for the L ∞ −norm of u in any compactly contained subset of Ω ⊂ R n is given in terms of the L r − norm of h and of the L p − norm of the gradient Du. See the details in the next Section, precisely in the statement of Theorem 2. In particular we obtain that any minimizer to the Dirichlet energy integral (1) is locally Lipschitz-continuous provided the coefficient a(x) belongs to W 1,r loc (Ω) for some r > n.
A motivation for the previous result, other that its intrinsic interest in the framework of regularity theory, also relies in the approximation procedure to pass from a-priori estimates to existence and regularity under general p, q−growth conditions. A place where this procedure has been used is the author's paper [15]; we plan to go back to this problem in the next future to explain with more details the use of Theorem 1 to get local Lipschitz continuity and regularity of solutions in a general context.

A-priori estimates.
Let Ω be an open set of R n . In the following, we say that u ∈ W 1,p loc (Ω) is a local minimizer of the energy integral (2) if for every open set Ω ′ compactly contained in Ω and for every ϕ ∈ W 1,p 0 (Ω ′ ). We assume that f : Ω × R n → [0, +∞) is a convex function with respect to the gradient variable ξ ∈ R n and it is strictly convex only at infinity. Precisely, the second derivatives of f are Carathéodory functions satisfying the growth conditions for a.e. x ∈ Ω and for all λ, ξ ∈ R n , with |ξ| ≥ M 0 , for some constants M 0 ≥ 0, Here h ∈ L r loc (Ω) for some r > n. We observe that here the ellipticity and growth assumptions hold only for large values of the gradient variable, i.e., we consider functionals which are uniformly convex only at infinity. In this context see [6], [17], [10] and recently [13], [14] and [9]. The Sobolev dependence on x recently has been considered in [22], [1] and for obstacle problems in [16].
We observe that we can transform f (x, ξ) into f (x, M 0 ξ), which satisfies the same assumptions for |ξ| ≥ 1 (with different constants depending on M 0 ). Therefore and without loss of generality, for clarity of exposition, we assume M 0 = 1.
Throughout the paper we will denote by B ρ and B R balls of radii ρ and R (ρ < R) compactly contained in Ω and with the same center.
In this section we assume the following supplementary assumptions on f which will be automatically satisfied in Section 3. Assume that f ∈ C 2 (Ω × R n ) and there exist two positive constants k, K such that ∀ξ ∈ R n , ∀x ∈ Ω The following a-priori estimate holds. (7) and (8) hold and let u ∈ W 1,p loc (Ω) be a local minimizer of the energy integral in (2). Then u is locally Lipschitz continuous in Ω and the following a-priori estimate holds
Proof. Let u ∈ W 1,p loc (Ω) be a local minimizer of (2). First of all we obtain an apriori estimate for the L ∞ −norm of the gradient of u which is independent of k and K, i.e. for every 0 < ρ < R ≤ ρ + 1, we prove that there exists a positive constant C depending only on n, r, p, M 1 , M 2 such that where Let us observe that since r > n where 2 * := 2n n−2 when n > 3 and 2 * is any fixed real number greater than 2m = 2r r−2 when n = 2. Thereforeβ > 0 in (11). The local minimizer u satisfies the following Euler first variation By (8), the technique of the difference quotients (see [19], [12], in particular [18], Chapter 8, Sections 8.1 and 8.2) gives Let η ∈ C 1 0 (Ω) and for any fixed for a suitable constant c Φ > 1. Here (a) + denotes the positive part of a ∈ R; in the following we denote Φ((|Du| − 1) + ) = Φ(|Du| − 1) + . We have then Let p ≥ 2, by (13) we have that |D 2 u| 2 ∈ L 1 loc (Ω) . Otherwise if 1 < p < 2, we use the fact that u ∈ W 1,∞ loc (Ω) to infer that there exists M = M (supp ϕ) such that |Du(x)| ≤ M for a.e. x ∈ supp ϕ. Now since p − 2 < 0 we have and by (13) we again get |D 2 u| 2 ∈ L 1 (supp ϕ). Therefore we can insert ϕ xi in the following second variation and we obtain In the following, constants will be denoted by C, regardless of their actual value. First of all, using the Cauchy-Schwarz inequality, the Young inequality and (7), we have Moreover, by the growth of f ξξ in (7), we obtain On the other hand, we have Consider now the fifth term in (16): Finally we need to estimate the sixth integral in (16). Let us observe that we want to consider growth conditions only at infinity, therefore we need to overcome the difficulty due to the presence of the term Φ ′ in this sixth integral. The idea is to use the same argument exploited in [13]. For any 0 < δ < 1 we have In {x : |Du(x)| ≥ 2} we get (|Du| − 1) + + δ ≤ 2(|Du| − 1) + and we estimate the last integral using the properties of Φ in (14) Therefore we finally obtain Now putting together all the previous estimates, for ε sufficiently small, we deduce that there exists a constant C depending on n, p, M 1 , M 2 such that At this point we set Φ(s) : It is easy to check that Φ satisfies (14) with c Φ = 2(1 + γ).

Let Φ h be a sequence of functions, with Φ h equal to Φ in [0, h] and extended to [h, +∞) with the constant value Φ(h). Then (18) holds for each Φ h and since Φ h and Φ ′
h converge monotonically to Φ and Φ ′ , by passing to the limit we have (10) with Φ defined in (19).

LOCAL LIPSCHITZ CONTINUITY WITH MILD ASSUMPTIONS 259
For ρ < R < ρ + 1 consider a function η with η = 1 in B ρ , supp η ⊂ B R and such that |Dη| ≤ 2 (R−ρ) ; we obtain where the constant C 0 only depends on n, r, p, M 1 , M 2 but is independent of γ. Fixed 0 < ρ 0 < R 0 ≤ ρ 0 + 1, we define the decreasing sequence of radii {ρ k } k≥1 We define recursively a sequence α k in the following way The following representation formula for α k can be easily proved by induction so that We rewrite (28) with R = ρ k , ρ = ρ k+1 , γ = α k m and we observe that By iteration, we deduce that there existβ andC such that such thatC which is finite because the series is convergent (α i from the representation formula (30) grows exponentially) and
The a-priori estimate (9) follows by the classical interpolation inequality for any s ≥ p, which permits to estimate the essential supremum of the gradient of the local minimizer in terms of its L p −norm.
Above we applied the following technical lemma whose proof is in [15].