Regularity of minimizers of vectorial integrals with p − q growth

Giovanni Cupini^{1}, Marcello Guidorzi^{2}, Elvira Mascolo^{1} ^{∗}

1Dipartimento di Matematica “U. Dini”, Università di Firenze Viale Morgagni 67/A, 50134 - Firenze, Italy. 2Dipartimento di Matematica, Università di Ferrara Via Machiavelli 35, 44100 - Ferrara, Italy.

Abstract

Local Lipschitz continuity of local minimizers of vectorial integrals

∫_{Ω} f(x, Du(x)) dx

is proved when f satisfies p−q growth condition and ξ ↦ f(x, ξ) is convex. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity.

Key words: Regularity, local minimizer, non standard growth.

AMS classification: 49N60, 35J20.

1 Introduction

In this paper we study the Lipschitz continuity of local minimizers of non homogeneous integrals

F(u; Ω) = ∫_{Ω} f(x, Du(x)) dx ,  (1.1)

where Ω ⊂ ℝ^{n} is a bounded open set, Du denotes the gradient of a vector valued function u : Ω → ℝ^{N} and n, N > 1. We say that u ∈ W_{loc}^{1,1}(Ω, ℝ^{N}) is a local minimizer of (1.1) if f(x, Du) ∈ L_{loc}^{1}(Ω) and F(u; spt φ) ≤ F(u + φ; spt φ) for any φ ∈ W^{1,1}(Ω, ℝ^{N}) with spt φ ⊂⊂ Ω. Assume that f : Ω × ℝ^{nN} → [0, +∞) is a Carathéodory function, convex with respect to the last variable, satisfying the following conditions (A1) there exist R > 0 and a function f̃ such that for a.e. x ∈ Ω and every

∗Corresponding author: Elvira Mascolo, Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 - Firenze, Italy. Fax number: 39-055-4222695, Phone number 39-055-4237126.

ξ ∈ ℝ^{nN} \ B_{R}(0)

f(x, ξ) = f̃(x, |ξ|) ,

(A2) f is p-uniformly convex at infinity, that is there exist p > 1 and ν > 0 such that, for a.e. x ∈ Ω and for every ξ_{1}, ξ_{2} ∈ ℝ^{nN} \ B_{R}(0) endpoints of a segment contained in the complement of B_{R}(0),

\frac{1}{2}[f(x, ξ_{1}) + f(x, ξ_{2})] ≥ f(x, \frac{ξ_{1} + ξ_{2}}{2} ) + ν(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2}, (1.2)

(A3) there exist L > 0 and q ≥ p such that

f(x, ξ) ≤ L(1 + |ξ|)^{q},

for a.e. x ∈ Ω and ξ ∈ ℝ^{nN},

(A4) for a.e. x ∈ Ω and every ξ ∈ R^{nN} \B_{R}(0) let D_{t}^{+}f̃(x, |ξ|) be the right-side derivative of f̃ with respect to t and denote D_{ξ_{i}^{α}}^{+}f(x, ξ) = D_{t}^{+}f̃(x, |ξ|)\frac{ξ_{i}^{α}}{|ξ|}. Then for every ξ ∈ R^{nN} \ B_{R}(0), the vector field x ↦ D_{ξ}^{+}f(x, ξ) is weakly differentiable and

|D_{x}D_{ξ}^{+}f(x, ξ)| ≤ L(1 + |ξ|)^{q−1}.

Assumption (A2) was introduced by Fonseca-Fusco-Marcellini [8] in order to study the existence of minimizers of some non convex variational problems. This condition, together with (A3), implies that there exist two positive constants c_{0}, c_{1} such that

−c_{0} + c_{1}|ξ|^{p} ≤ f(x, ξ) ≤ L(1 + |ξ|)^{q}

(see Theorem 2.5 (i) below), thus f satisfies the so called p−q growth condition.

The main result of this paper is the following

Theorem 1.1 Let u be a local minimizer of (1.1), whose integrand f satisfies the assumptions (A1)–(A4), 1 < p ≤ q < p(n+1)/n. Then u is locally Lipschitz continuous and for all B_{r}(x_{0}) ⊂⊂ Ω

sup_{B_{r/4}(x_{0})} |Du| ≤ c [∫_{B_{r}(x_{0})}(1 + f(x, Du)) dx]^{β} ,  (1.3)

where c = c(n, p, q, L, R, ν), and β = β(n, p, q).

When f is not convex, the previous theorem still holds provided

sc^{−}F(w; Ω) = ∫_{Ω} f^{∗∗}(x, Dw) dx ,  (1.4)

where sc^{−}F is the relaxed functional of F and ξ ↦ f^{∗∗}(x, ξ) is the convex envelope of f(x, ·). In fact by Theorem 2.5 (iv) f^{∗∗} fulfills (A1)–(A4) with suitable constants ν and R. We refer to [4] for some cases where (1.4) holds. The result stated in Theorem 1.1 covers some interesting models in fluids mechanics and in non-linear elasticity, see [15, 22] and applies also to weak solutions of non-linear elliptic systems in divergence form

∑_{i=1}^{n} D_{i}a_{i}^{α}(x, Du) = 0 for α = 1, …, N

where a_{i}^{α}(x, ξ) = f_{ξ_{i}^{α}}(x, ξ). Notice that by (A2) a_{i}^{α} may be degenerate elliptic for small deformations. It is well known that, if N > 1, generally only partial regularity of local minimizers of (1.1) may be achieved. However, starting from the seminal paper by Uhlenbeck [23], everywhere regularity results were proved under standard growth conditions, p = q, provided f = f(|ξ|): in particular Giaquinta-Modica in [13] (p ≥ 2) and Acerbi-Fusco in [1] (1 < p < 2) assume that f is of class C^{2} and

ν(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} ≤ ⟨D^{2}f(ξ)λ, λ⟩ ≤ L(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2}  (1.5)

for all ξ, λ ∈ ℝ^{nN}. Using an approximation argument introduced by Fonseca-Fusco [7], Lipschitz continuity of local minimizers can still be achieved even if f is only continuous, see Fusco-Mingione-Trombetti [9]. The ellipticity condition in (1.5) is replaced by the p-uniform convexity assumption, that is inequality (1.2) holds for any x ∈ Ω and ξ_{1}, ξ_{2} ∈ ℝ^{nN}. The non standard context p < q is more recent. Firstly studied by Marcellini [17], various contributions to the subject have been provided in these last years, see [18, 19] and [6] for an extensive list of references. The vectorial homogeneous framework was considered by Marcellini [20] when f is smooth, and by Esposito-Leonetti-Mingione [5] when f is continuous. It is well known that in order to get regularity a restriction between p and q need to be imposed, namely q ≤ c(n)p, with c(n) close to 1, see [15] and [11] for counterexamples. In the non homogeneous case the bound revealing the sharpness of our result can be found. In [5] Esposito-Leonetti-Mingione provide an example from which it can be deduced that Theorem 1.1 does not hold if q > p\frac{n+1}{n}, since local minimizers may not belong to W_{loc}^{1,q}(Ω, ℝ^{N}). Also some regularity conditions with respect to x are needed. A functional F exists, with f only measurable with respect to x, such that its minimizers do not belong to W^{1,q}(Ω) (see Zhikov [24]). On the other hand, when the same functional satisfies a suitable continuity assumption with respect to x, the minimizers are regular, see Coscia-Mingione [3]. Recently, Mascolo-Migliorini [21] proved a Lipschitz continuity result for minimizers of smooth and strictly convex functionals, with p − q growth and p ≥ 2. There, also functionals with exponential growth are considered.

Let us briefly discuss the techniques used to prove Theorem 1.1. We construct a sequence of regular functions f_{kh} satisfying (A1)–(A4), p-growth condition, with constants independent of h, and (1.5), with suitable constants depending on h. Fixed a ball B_{r}(x_{0}) ⊂⊂ Ω, we consider the following sequence of variational problems

min {∫_{B_{r}(x_{0})} f_{kh}(x, Dw) dx : w ∈ u + W_{0}^{1,p}(B_{r}(x_{0}), ℝ^{N})} ,

where u is a local minimizer of (1.1). For the corresponding minimizers u_{kh} we prove a uniform (in k and h) estimate for the L^{∞}-norm of the gradient. The passage to the limit, with respect to h and k, is not influenced by the behaviour of f on bounded subsets of Ω×ℝ^{nN}. This and a suitable comparison argument, introduced in [8], permit to transfer the regularity properties of the minimizers u_{kh} to the local minimizer u. The paper is organized as follows: in Section 2 we study some properties of functions uniformly convex at infinity. Section 3 is devoted to the proof of the a priori estimates for minimizers of regular functionals, while in Section 4 we construct the approximating variational problems and we prove Theorem 1.1.

2 Uniformly convex functions

In this section we prove some properties satisfied by functions uniformly convex at infinity. For the sake of clarity we first consider the case when f : ℝ^{m} → [0, +∞) is a continuous function, f = f(ξ), satisfying the following assumption

(UC) there exist p > 1 and ν, R > 0 such that for every ξ_{1}, ξ_{2} ∈ ℝ^{m} \ B_{R}(0), endpoints of a segment contained in the complement of B_{R}(0),

\frac{1}{2}[f(ξ_{1}) + f(ξ_{2})] ≥ f(\frac{ξ_{1} + ξ_{2}}{2} ) + ν(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2}. (2.1)

Some of the results are stated in [8]. Here we give alternative proofs, taking into account neither the growth of f nor its behaviour in B_{R}(0).

When f is smooth, uniform ellipticity and uniform convexity are equivalent conditions (see [8] for the proof).

Lemma 2.1 Let f : ℝ^{m} → [0, +∞) be of class C^{2}. Then f satisfies (UC) if and only if there exists c(ν) > 0 such that for all ξ ∈ ℝ^{m} \ B_{R}(0) and λ ∈ ℝ^{m}

⟨D_{ξξ}f(ξ)λ, λ⟩ ≥ c(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2}.  (2.2)

Moreover, (2.1) holds for all ξ_{1}, ξ_{2} in ℝ^{m} if and only if (2.2) holds for all ξ and λ in ℝ^{m}.

Now, we prove that the p-uniform convexity at infinity implies that f is pcoercive.

Lemma 2.2 Let f : ℝ^{m} → [0, +∞) satisfy (UC). Then there exist two positive constants c_{0}(p, ν, R, max_{∂B_{R}} f) and c_{1}(ν) such that

f(ξ) ≥ −c_{0} + c_{1}|ξ|^{p} .  (2.3)

Proof. For every ξ ∈ ℝ^{m} \ B_{R}(0) define ξ_{0} = Rξ/|ξ|. From (UC) it follows

\frac{1}{2}[f(ξ) + f(ξ_{0})] ≥ f(\frac{ξ + ξ_{0}}{2} ) + ν(1 + |ξ|^{2} + |ξ_{0}|^{2})^{\frac{p−2}{2}} |ξ − ξ_{0}|^{2} .

On the other hand, Young inequality yields

(1 + |ξ|^{2} + |ξ_{0}|^{2})^{\frac{p−2}{2}} |ξ − ξ_{0}|^{2} ≥ \frac{1}{2}|ξ|^{p} − c(p, R).

Since f ≥ 0, we get

f(ξ) ≥ \frac{ν}{2} |ξ|^{p} − 2c(p, R)ν − max_{∂B_{R}} f

and (2.3) follows. ■

The next proposition describes the “splitting” property of functions p-uniformly convex at infinity.

Proposition 2.3 Let f : ℝ^{m} → [0, +∞). (UC) is equivalent to

(S) there exist c_{0} and c_{1} positive constants and a function g : ℝ^{m} → [−c_{0}, +∞) such that for every ξ ∈ ℝ^{m},

f(ξ) = c_{1}(1 + |ξ|^{2})^{\frac{p}{2}} + g(ξ)

and for every ξ_{1}, ξ_{2} ∈ ℝ^{m} such that [ξ_{1}, ξ_{2}] ⊆ ℝ^{m} \ B_{R}(0)

\frac{1}{2}[g(ξ_{1}) + g(ξ_{2})] ≥ g(\frac{ξ_{1} + ξ_{2}}{2} ) .

Proof. (UC) ⇒ (S). It is easy to check that there exists c(p) > 0 such that for all ξ_{1}, ξ_{2} ∈ ℝ^{m}

(1 + |ξ_{1}|^{2})^{\frac{p}{2}} + (1 + |ξ_{2}|^{2})^{\frac{p}{2}}
≤ 2 (1 + |\frac{ξ_{1} + ξ_{2}}{2} |^{2})^{\frac{p}{2}} + c(p) (1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2}.  (2.4)

Moreover, from Lemma 2.2,

f(ξ) ≥ −c_{0} + c̃_{1}(1 + |ξ|^{2})^{\frac{p}{2}}

for some positive constants c_{0} and c̃_{1}. Let c_{1} = min{c̃_{1}, 2ν/c(p)}. The function

g(ξ) := f(ξ) − c_{1}(1 + |ξ|^{2})^{\frac{p}{2}}

is continuous and g ≥ −c_{0}. (UC) and (2.4) imply

\frac{1}{2}[g(ξ_{1}) + g(ξ_{2})] ≥ g (\frac{ξ_{1} + ξ_{2}}{2} ) + (ν − c(p)\frac{c_{1}}{2} ) (1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2},

for all ξ_{1}, ξ_{2} with [ξ_{1}, ξ_{2}] ⊆ ℝ^{m} \ B_{R}(0).

(S) ⇒ (UC). From Lemma 2.1, the function (1 + |ξ|^{2})^{\frac{p}{2}} is p-uniformly convex in ℝ^{m}, so (S) implies (UC). ■

Proposition 2.4 Let f : ℝ^{m} → [0, +∞) satisfy (UC). Then there exist ν_{0}, R_{0} > 0 depending only on p, ν, R and max_{∂B_{R}(0)} f, such that for any ξ ∈ ℝ^{m} \ B_{R_{0}}(0) there exists q_{ξ} ∈ ℝ^{m}, such that

f(η) ≥ f(ξ) + ⟨q_{ξ}, η − ξ⟩ + ν_{0}(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}  (2.5)

for all η ∈ ℝ^{m}. Moreover, if f^{∗∗} is the convex envelope of f, then f = f^{∗∗} in ℝ^{m} \ B_{R_{0}}(0).

Proof. We divide the proof into two steps.

Step 1. Assume that f is a convex function. For 0 < ε < 1, let us define f_{ε} = ρ_{ε} ∗ f, with ρ_{ε}(η) = ε^{−m}ρ(η/ε), where ρ is a positive radially symmetric mollifier with support in B = B_{1}(0), such that ∫_{B} ρ = 1. Let [ξ_{1}, ξ_{2}] ⊆ ℝ^{m} \ B_{R+1} and ξ_{0} = \frac{ξ_{1}+ξ_{2}}{2} . Since

∫_{B}(1 + |ξ_{1} + εη|^{2} + |ξ_{2} + εη|^{2})^{\frac{p−2}{2}} dη ≥ c(p)(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} ,

then

\frac{1}{2} [f_{ε}(ξ_{1}) + f_{ε}(ξ_{2})] ≥ f_{ε}(ξ_{0}) + c(p, ν)(1 + |ξ_{1}|^{2} + |ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1} − ξ_{2}|^{2}.

Thus, f_{ε} is p-uniformly convex at infinity and, by Lemma 2.1, it follows that for all |ξ| ≥ R + 1 and λ ∈ ℝ^{m}

⟨D_{ξξ}f_{ε}(ξ)λ, λ⟩ ≥ c(p, ν)(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2}.  (2.6)

Fix ξ ∈ ℝ^{m} \ B_{R_{0}}, with R_{0} = 2(R + 1). For all η ∈ ℝ^{m}, define A_{ξ,η} = {t : |ξ + t(η − ξ)| > R + 1}. The convexity of f_{ε} implies

\begin{matrix}f_{ε}( & η) = f_{ε}(ξ) + ⟨Df_{ε}(ξ), η − ξ⟩ +∫_{0}^{1}(1 − t)⟨D^{2}f_{ε}(ξ + t(η − ξ))(η − ξ), η − ξ⟩ dt \\ ≥ f_{ε}(ξ) + ⟨Df_{ε}(ξ), η − ξ⟩ + ∫_{A_{ξ,η}}(1 − t)⟨D^{2}f_{ε}(ξ + t(η − ξ))(η − ξ), η − ξ⟩ dt. \end{matrix} (2.7)

Let us prove that (2.5) holds for all η ∈ ℝ^{m} such that |ξ| > |η|. When 0 < t < \frac{1}{4}

|ξ + t(η − ξ)| ≥ (1 − t)|ξ| − t|η| > (1 − 2t)R_{0} > R + 1 ,  (2.8)

moreover there exists c_{1} > 0 such that

c_{1}(|ξ|^{2} + |η|^{2}) ≤ |ξ + t(η − ξ)|^{2} ≤ 2(|ξ|^{2} + |η|^{2}) .  (2.9)

From (2.8), (0, \frac{1}{4}) ⊂ A_{ξ,η} and, by (2.6) and (2.9),

∫_{A_{ξ,η}}(1 − t)⟨D^{2}f_{ε}(ξ + t(η − ξ))(η − ξ), η − ξ⟩ dt

≥ c(p, ν) ∫0^{\frac{1}{4}} (1 − t)(1 + |ξ + t(η − ξ)|^{2})^{\frac{p−2}{2}} |η − ξ|^{2} dt
≥ c(p, ν)(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |η − ξ|^{2}.  (2.10)

Collecting (2.7) and (2.10), we get

f_{ε}(η) ≥ f_{ε}(ξ) + ⟨Df_{ε}(ξ), η − ξ⟩ + c(p, ν)(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}. (2.11)

Since f is locally Lipschitz continuous, then for all λ ∈ B_{1}(ξ)

|f_{ε}(λ) − f_{ε}(ξ)| ≤ ∫_{B} |f(λ + εy) − f(ξ + εy)|ρ(y) dy ≤ c(ξ)|λ − ξ| ,

so that (D_{ε}f(ξ))_{ε} is bounded with respect to ε. Thus, up to a subsequence, Df_{ε}(ξ) converges to a certain q_{ξ} ∈ ℝ^{m} when ε → 0. (2.11) implies (2.5) when |ξ| > |η|. Let now η ∈ ℝ^{m} be such that |ξ| ≤ |η|. If t > \frac{3}{4}, then |ξ + t(η − ξ)| is greater than R + 1. Hence (\frac{3}{4}, 1) ⊂ A_{ξ,η} and

\begin{aligned}& ∫_{A_{ξ,η}}(1 − t)⟨D^{2}f_{ε}(ξ + t(η − ξ))(η − ξ), η − ξ⟩ dt \\ ≥ & c(p, ν) ∫ 1_{\frac{3}{4}}(1 − t)(1 + |ξ + t(η − ξ)|^{2})^{\frac{p−2}{2}} |η − ξ|^{2} dt . \end{aligned}

Reasoning as above, we get (2.5).

Step 2. Let now f be a continuous function satisfying (UC). Set M = max_{∂B_{R}} f and define

h(ξ) := \{ \begin{matrix}M & if |ξ| ≤ R , \\ max\{M, f(ξ)\} & if |ξ| > R . \end{matrix}

We claim that h is convex, i.e. given ξ_{1}, ξ_{2} ∈ ℝ^{m} and ξ_{0} = \frac{ξ_{1}+ξ_{2}}{2} , then

h(ξ_{0}) ≤ \frac{1}{2}[h(ξ_{1}) + h(ξ_{2})].  (2.12)

If ξ_{0} ∈ B_{R}, (2.12) is trivial being h ≥ M. Let |ξ_{0}| ≥ R. If [ξ_{1}, ξ_{2}] is subset of ℝ^{m} \ B_{R}, then (2.12) follows from (UC) and h ≥ f; if not, either the segment [ξ_{0}, ξ_{1}] ⊂ ℝ^{m} \ B_{R} or else [ξ_{0}, ξ_{2}] ⊂ ℝ^{m} \ B_{R}. Suppose the last one. Clearly, there exists γ ∈ ∂B_{R} such that ξ_{0} ∈ [γ, ξ_{2}] ⊂ [ξ_{1}, ξ_{2}]. Let τ ∈ (0, 1/2] be such that λ = τγ + (1 − τ)ξ_{2}. Being h convex on [γ, ξ_{2}] and M = h(γ) = min_{[γ,ξ_{2}]} h, we get

h(ξ_{0}) ≤ h(\frac{γ + ξ_{2}}{2} ) ≤ \frac{1}{2}[M + h(ξ_{2})],

so (2.12) follows. By Lemma 2.2, there exists R_{1} ≥ R, depending on p, ν, R and M, such that h(ξ) = f(ξ) when |ξ| ≥ R_{1}, which implies that h is uniformly convex in ℝ^{m} \ B_{R_{1}}(0). From step 1, there exist R_{2} ≥ R_{1}, ν_{0}(p, ν) > 0 and a vector q_{ξ} such that for |ξ| > R_{2} and η ∈ ℝ^{m}

\begin{aligned}h(η) & ≥ h(ξ) + ⟨q_{ξ}, η − ξ⟩ + ν_{0}(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} \\ & = f(ξ) + ⟨q_{ξ}, η − ξ⟩ + ν_{0}(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}, \end{aligned}  (2.13)

therefore (2.5) holds, when |η| > R_{1}. Let |η| ≤ R_{1}. Since for some c_{1}, c_{2} > 0

(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} ≥ c_{1}(p, ν)|ξ|^{p} − c_{2}(p, R_{1}, ν),  (2.14)

taking into account (2.13), we have

h(η) − c_{1}|ξ|^{p} + c_{2} ≥ f(ξ) + ⟨q_{ξ}, η − ξ⟩ + \frac{ν_{0}}{2} (1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}.

We choose R_{0} ≥ R_{2} such that, when |ξ| > R_{0}, for every η ∈ \overline{B}_{R_{1}}(0) is h(η) − c_{1}|ξ|^{p} + c_{2} < 0. Since f ≥ 0, (2.5) follows.

Fixed ξ ∈ ℝ^{m} \ B_{R_{0}}(0), for any η ∈ ℝ^{m} define

v_{ξ}(η) := f(ξ) + ⟨q_{ξ}, η − ξ⟩.

v_{ξ} is convex and, from (2.5), v_{ξ}(η) ≤ f(η). Then v_{ξ}(η) ≤ f^{∗∗}(η) ≤ f(η) where f^{∗∗} denotes the convex envelope of f. Therefore f(ξ) = f^{∗∗}(ξ). ■

The previous results still hold when f is not homogeneous and satifies (A3). The proof is straightforward and is left to the reader.

Theorem 2.5 Let f : Ω × ℝ^{nN} → [0, +∞), f = f(x, ξ), be a Carathéodory function. Then (A2) and (A3) imply

(i) there exist c_{0}(p, q, ν, R, L) and c_{1}(ν) such that for a.e. x ∈ Ω and every ξ ∈ ℝ^{nN}

f(x, ξ) ≥ −c_{0} + c_{1}|ξ|^{p},

(ii) there exist c_{0}(p, q, ν, R, L), c_{1}(p, ν) > 0 and a Carathéodory function g : Ω × ℝ^{nN} → [−c_{0}, +∞) such that for a.e. x ∈ Ω and every ξ ∈ ℝ^{nN}

f(x, ξ) = c_{1}(1 + |ξ|^{2})^{\frac{p}{2}} + g(x, ξ).  (2.15)

Moreover, for a.e. x ∈ Ω and every ξ_{1}, ξ_{2} ∈ ℝ^{nN} such that [ξ_{1}, ξ_{2}] ⊆ ℝ^{nN} \ B_{R}(0)

\frac{1}{2}[g(x, ξ_{1}) + g(x, ξ_{2})] ≥ g(x, \frac{ξ_{1} + ξ_{2}}{2} ) ,

(iii) there exist R_{0}, ν_{0} > 0 depending only on p, q, ν, R and L, such that for a.e. x ∈ Ω and every ξ ∈ ℝ^{nN} \ B_{R_{0}}(0) there exists q_{ξ}(x) ∈ ℝ^{nN}, |q_{ξ}(x)| ≤ c(q, L)(1 + |ξ|)^{q−1}, such that for all η ∈ ℝ^{nN}

f(x, η) ≥ f(x, ξ) + ⟨q_{ξ}(x), η − ξ⟩ + ν_{0}(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}.

Moreover, if ξ ↦ f(x, ξ) is C^{1}(ℝ^{nN} \ B_{R}(0)), then q_{ξ}(x) = D_{ξ}f(x, ξ),

(iv) if ξ ↦ f^{∗∗}(x, ξ) is the convex envelope of ξ ↦ f(x, ξ), then f = f^{∗∗} in Ω × (ℝ^{nN} \ B_{R_{0}}(0)).

Moreover, if f satisfies (ii) for some c_{0}, c_{1} and R, then (A2) holds, with the same R and with ν depending only on p and c_{1}.

Theorem 2.5 (iii) enables us to compare different local minimizers of functionals whose densities satisfy the uniform convexity at infinity. The following proposition is essentially contained in the proof of Theorem 2.7 in [8]. For the sake of completeness we give a detailed proof.

Lemma 2.6 Let u be a local minimizer of

F(w; Ω) = ∫_{Ω} f(x, Dw(x)) dx ,

where f : Ω×ℝ^{nN} → [0, +∞) is a Carathéodory function, convex with respect to the last variable, satisfying (A2) and (A3). Suppose that there exist B_{r}(x_{0}) ⊂⊂ Ω and v ∈ u + W_{0}^{1,1}(B_{r}(x_{0}), ℝ^{N}) such that F(u; B_{r}(x_{0})) = F(v; B_{r}(x_{0})). Then there exists R_{0}(p, q, ν, R, L) > R such that the Lebesgue measure of the set

{x ∈ B_{r}(x_{0}) : |Du(x) + Dv(x)| > 2R_{0} and |Du(x) − Dv(x)| > 0}

is zero.

Proof. From Theorem 2.5 (iii) there exist R_{0}, ν_{0} > 0 such that for a.e. x ∈ Ω and every ξ ∈ ℝ^{nN} \ B_{R_{0}}(0)

f(x, η) ≥ f(x, ξ) + ⟨q_{ξ}(x), η − ξ⟩ + ν_{0}(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2}  (2.16)

for some q_{ξ}(x) ∈ ℝ^{nN} and for all η ∈ ℝ^{nN}. Set A = {x ∈ B_{r}(x_{0}) : |Du + Dv| > 2R_{0}}. For a.e. x ∈ A consider (2.16) with ξ = (Du(x) + Dv(x))/2 and η = Du(x),

\begin{matrix}f(x, Du) − f(x, \frac{Du + Dv}{2}) ≥ ⟨q_{ξ}(x), \frac{Du − Dv}{2} ⟩ \\ + & ν_{0} (1 + |\frac{Du + Dv}{2} |^{2} + |Du|^{2})^{\frac{p−2}{2}} |\frac{Du − Dv}{2} |^{2} \end{matrix}  (2.17)

and (2.16) applied with η = Dv(x)

\begin{matrix}f(x, Dv) − f(x, \frac{Du + Dv}{2}) ≥ ⟨q_{ξ}(x), \frac{Dv − Du}{2}⟩ \\ + & ν_{0} (1 + |\frac{Du + Dv}{2} |^{2} + |Dv|^{2})^{\frac{p−2}{2}} |\frac{Du − Dv}{2} |^{2} ; \end{matrix}  (2.18)

thus, summing (2.17) and (2.18) and integrating on A, we get

\begin{matrix}∫_{A} [f(x, Du) + f(x, Dv)] dx ≥ 2 ∫_{A} f (x, \frac{Du + Dv}{2} ) dx \\ + & ν_{0} ∫_{A} (1 + |\frac{Du + Dv}{2} |^{2} + |Du|^{2})^{\frac{p−2}{2}} |\frac{Du − Dv}{2} |^{2} dx \\ + & ν_{0} ∫_{A} (1 + |\frac{Du + Dv}{2} |^{2} + |Dv|^{2})^{\frac{p−2}{2}} |\frac{Du − Dv}{2} |^{2} dx. \end{matrix}  (2.19)

The convexity of f with respect to ξ and (2.19) give

F (\frac{u + v}{2} ; B_{r}) − \frac{1}{2}F(u; B_{r}) − \frac{1}{2}F(v; B_{r})

≤ − \frac{ν_{0}}{2} ∫_{A}(1 + |\frac{Du + Dv}{2} |^{2}+ |Du|^{2})^{\frac{p−2}{2}} |\frac{Du − Dv}{2} |^{2} dx

−\frac{ν_{0}}{2} ∫_{A}(1 + |\frac{Du + Dv}{2} |^{2}+ |Dv|^{2})^{\frac{p−2}{2}} |\frac{Du − Dv}{2} |^{2} dx

and since F(u; B_{r}) = F(v; B_{r}) the thesis follows. ■

3 A Priori estimates

In order to prove Theorem 1.1, we prove some regularity results for minimizers of functionals with smooth integrands.

Let us consider

F(u; Ω) = ∫_{Ω} f(x, Du) dx  (3.1)

where f : Ω × ℝ^{nN} → [0, +∞), f = f(x, ξ), is convex with respect to ξ, satisfies (A1)–(A4) and the supplementary assumptions

(H1) f is in C^{2}(Ω × ℝ^{nN}),

(H2) there exists K > 0 such that for every (x, ξ) ∈ Ω × ℝ^{nN}

f(x, ξ) ≤ K(1 + |ξ|^{2})^{\frac{p}{2}} ,

(H3) there exist ϵ_{0} and Λ positive constants such that for every x ∈ Ω and for every ξ, λ ∈ ℝ^{nN}

ϵ_{0}(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} ≤ ⟨D_{ξξ}f(x, ξ)λ, λ⟩ ≤ Λ(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2},

(H4) for every (x, ξ) ∈ Ω × ℝ^{nN}

|D_{ξx}f(x, ξ)| ≤ K(1 + |ξ|^{2})^{\frac{p−1}{2}} .

The following result holds:

Proposition 3.1 Let u be a local minimizer of (3.1), where f satisfies (A1), (A2) and (H1)–(H4). Fixed B_{r}(x_{0}) ⊂⊂ Ω, there exist C = C(n, p, ν, R, K, r) > 0 such that

sup_{B_{r/4}(x_{0})} |Du| ≤ C [∫_{B_{r}(x_{0})}(1 + |Du|^{2})^{\frac{p}{2}} dx]^{\frac{1}{p}} .  (3.2)

We postpone the proof of Proposition 3.1 to some preliminary results. Next lemma can be proved as Lemma 4.3 in [9]. From (A1), we define \overline{f} such that, for a.e. x ∈ Ω and for every ξ ∈ R^{nN} \ B_{R}(0), f(x, ξ) = \overline{f}(x, |ξ|^{2}).

Lemma 3.2 Let f : Ω × ℝ^{nN} → [0, +∞) be of class C^{2} and let (A1) and (A2) hold. Then for all x ∈ Ω, ξ ∈ ℝ^{nN} \ B_{R}(0) and λ ∈ ℝ^{n}

∑_{α,s,i} 4\overline{f}_{tt}(x, |ξ|^{2})ξ_{s}^{α}ξ_{i}^{α}λ_{s}λ_{i} + 2\overline{f}_{t}(x, |ξ|^{2})|λ|^{2} ≥ c(ν)(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} .

The following result extends to our case some integrability properties for local minimizers proved in [13] (p ≥ 2) and in [1] (1 < p < 2) when f = f(ξ). We omit the proof which closely follows that of Lemma 2.5 in [1].

Lemma 3.3 Let u be a local minimizer of (3.1), with f satisfying (H1)–(H4). Then u ∈ W_{loc}^{2,a}(Ω), with a = min\{2, p\}, and (1 + |Du|^{2})^{\frac{p−2}{2}} |D^{2}u|^{2} ∈ L_{loc}^{1}(Ω).

Next lemma concernes the validity of Euler systems. It generalizes Lemma 4.1 in [5] to non homogeneous energy densities, with a similar proof that we omit.

Lemma 3.4 Let u be a local minimizer of (3.1), with f satisfying (H1)–(H4). Let B_{r}(x_{0}) ⊂⊂ Ω and suppose that Θ is a W^{1,1}(B_{r}(x_{0}), ℝ^{N}) function with compact support in B_{r}, such that

D_{ξ_{i}^{α}}f(x, Du)Θ^{α}, D_{i}(D_{ξ_{i}^{α}}f(x, Du))Θ^{α}, D_{ξ_{i}^{α}}f(x, Du)D_{i}Θ^{α}

belong to L^{1}(B_{r}), for all i = 1, ..., n, α = 1, ..., N. Then, for all α,

∑_{i} ∫_{B_{r}(x_{0})} D_{ξ_{i}^{α}}f(x, Du)D_{i}Θ^{α} dx = 0 .

Proof of Proposition 3.1. We divide it into two steps.

Step 1. Let us consider a cut-off function η ∈ C_{c}^{1}(B_{r}), 0 ≤ η ≤ 1. We prove that there exists c depending on p, ν, K such that for all δ > −1 there exists

\begin{matrix}∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2} +δ−1}|D(|Du|^{2} − R^{2})_{+}|^{2}η^{2} dx \\ ≤ & c\frac{max\{1, 1 + δ\}}{min\{1, 1 + δ\}} ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2} +δ+1}(η^{2} + |Dη|^{2}) dx. \end{matrix}  (3.3)

For any M > 0 let γ_{M} ∈ C^{∞}([0, +∞), [0, 1]) be such that γ_{M} ≡ 1 in [0, M], γ_{M}(t) = 0 if t ≥ 2M, |γ′_{M}(t)| ≤ 2/M. Fixed δ > −1 let us define Φ : [0, +∞) → [0, +∞),

Φ(t) := (1 + R^{2} + t)^{δ}t .  (3.4)

By a well known property of summable functions there exists an increasing and divergent sequence (a_{h}) in ℝ_{+} such that the Lebesgue measure of {x : (|Du(x)|^{2} − R^{2})_{+} = a_{h}} is zero. For every h ∈ ℕ let us define

Φ_{h}(t) := \{ \begin{matrix}Φ(t) & if t < a_{h}, \\ Φ(a_{h}) & otherwise . \end{matrix}

Notice that, for all h, Φ_{h}(0) = 0, Φ_{h} and Φ′_{h} are nonnegative, bounded and that if h goes to infinity they monotonically converge to Φ and Φ′, respectively. Recall that the difference quotient △_{s,h} in the direction e_{s} is defined by

△_{s,h}v(x) := \frac{v(x + he_{s}) − v(x)}{h} .

For any h, M, s = 1, ..., n, α = 1, ..., N and k > 0, small enough, define

Ψ_{h,M,s,k}^{α}(x) := γ_{M}(|Du(x)| + |Du(x + ke_{s})|)Φ_{h}((|Du(x)|^{2} − R^{2})_{+})D_{s}u^{α}(x) and

Θ_{h,M,s,k}^{α}(x) := η^{2}(x)△_{s,−k}Ψ_{h,M,s,k}^{α}(x) .

By Lemma 3.4, for all α

∑_{i} ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)D_{i}Θ_{h,M,s,k}^{α}(x) dx = 0 .  (3.5)

For simplicity we shall write γ_{M,k}, γ_{M}, Φ_{h} in place of γ_{M}(|Du(x)| + |Du(x + ke_{s})|), γ_{M}(2|Du|) and Φ_{h}((|Du(x)|^{2} −R^{2})_{+}), respectively. By (3.5), integrating by parts and summing on α and s we get

∑_{i,α,s} ∫_{B_{r}} △_{s,k}[D_{ξ_{i}^{α}}f(x, Du)η^{2}]D_{i}Ψ_{h,M,s,k}^{α} dx

= ∑_{i,α,s} 2 ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)ηD_{i}η△_{s,−k}Ψ_{h,M,s,k}^{α} dx . (3.6)

From now on, we shall use the Einstein’s convention for repeated indexes. Letting k go to 0 in (3.6) we get

∫_{B_{r}} D_{ξ_{i}^{α}ξ_{j}^{β}}f(x, Du)D_{sj}u^{β}η^{2}D_{i}[γ_{M}Φ_{h}D_{s}u^{α}] dx
= − ∫_{B_{r}} D_{x_{s}ξ_{i}^{α}}f(x, Du)η^{2}D_{i}[γ_{M}Φ_{h}D_{s}u^{α}] dx
−2 ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)ηD_{s}ηD_{i}[γ_{M}Φ_{h}D_{s}u^{α}] dx
+2 ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)ηD_{i}ηD_{s}[γ_{M}Φ_{h}D_{s}u^{α}] dx

so that

∫_{B_{r}} D_{ξ_{i}^{α}ξ_{j}^{β}}f(x, Du)D_{sj}u^{β}η^{2}γ_{M}
· [Φ_{h}D_{is}u^{α} + Φ′_{h}D_{i}((|Du|^{2} − R^{2})_{+})D_{s}u^{α}] dx
= −2 ∫_{B_{r}} D_{ξ_{i}^{α}ξ_{j}^{β}}f(x, Du)D_{sj}u^{β}γ′_{M}D_{i}(|Du|)Φ_{h}D_{s}u^{α}η^{2} dx
− 2 ∫_{B_{r}} D_{x_{s}ξ_{i}^{α}}f(x, Du)Φ_{h}γ′_{M}D_{i}(|Du|)D_{s}u^{α}η^{2} dx
− ∫_{B_{r}} D_{x_{s}ξ_{i}^{α}}f(x, Du)γ_{M}η^{2}[Φ′_{h}D_{i}((|Du|^{2} − R^{2})_{+})D_{s}u^{α} + Φ_{h}D_{is}u^{α}] dx
− 2 ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)γ_{M}ηD_{s}η[Φ_{h}D_{is}u^{α} + Φ′_{h}D_{i}((|Du|^{2} − R^{2})_{+})D_{s}u^{α}] dx
− 2 ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)Φ_{h}γ′_{M}D_{i}(|Du|)D_{s}u^{α}ηD_{s}η dx .
+ 2 ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)γ_{M}ηD_{i}η[Φ_{h}D_{ss}u^{α} + Φ′_{h}D_{s}((|Du|^{2} − R^{2})_{+})D_{s}u^{α}] dx
+ 2 ∫_{B_{r}} D_{ξ_{i}^{α}}f(x, Du)Φ_{h}γ′_{M}D_{s}u^{α}D_{s}(|Du|)ηD_{i}η dx .  (3.7)

Recalling that Φ_{h} stands for Φ_{h}((|Du(x)|^{2} − R^{2})_{+}) and that Φ_{h}(0) = 0, (A2) and Lemma 2.1 imply

\begin{aligned}& ∫_{B_{r}} D_{ξ_{i}^{α}ξ_{j}^{β}}f(x, Du)D_{js}u^{β}D_{is}u^{α}Φ_{h}γ_{M}η^{2} dx \\ ≥ & c(ν) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D^{2}u|^{2}Φ_{h}γ_{M}η^{2}dx . \end{aligned}

On the other hand, since

D_{ξ_{i}^{α}ξ_{j}^{β}}f(x, ξ) = 4\overline{f}_{tt}(x, |ξ|^{2})ξ_{i}^{α}ξ_{j}^{β} + 2\overline{f}_{t}(x, |ξ|^{2})δ^{ij}δ_{αβ} ,

by Lemma 3.2 it follows

∫_{B_{r}} D_{ξ_{i}^{α}ξ_{j}^{β}}f(x, Du)D_{s,j}u^{β}D_{i}((|Du|^{2} − R^{2})_{+})D_{s}u^{α}Φ′_{h}γ_{M}η^{2} dx
= 2 ∫_{B_{r}} \overline{f}_{tt}(x, |Du|^{2})D_{s}u^{α}D_{i}u^{α}D_{s}(|Du|^{2})D_{i}(|Du|^{2} − R^{2})_{+}Φ′_{h}γ_{M}η^{2} dx
+ ∫_{B_{r}} \overline{f}_{t}(x, |Du|^{2})D_{i}(|Du|^{2})D_{i}(|Du|^{2} − R^{2})_{+}Φ′_{h}γ_{M}η^{2} dx
≥ c(ν) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D(|Du|^{2} − R^{2})_{+}|^{2}Φ′_{h}γ_{M}η^{2} dx .  (3.8)

In the following we denote by c any constant which may take different values from line to line.

(H2) and the convexity of f with respect to ξ give

|D_{ξ}f(x, ξ)| ≤ c(K)(1 + |ξ|)^{p−1}  (3.9)

for all ξ in ℝ^{nN}. (H3), (H4), (3.7)–(3.9) imply

\begin{matrix}∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D^{2}u|^{2}Φ_{h}γ_{M}η^{2}dx \\ + ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D(|Du|^{2} − R^{2})_{+}|^{2}Φ′_{h}γ_{M}η^{2} dx \\ ≤ & c(p, ν, K, Λ) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−1}{2}} |D^{2}u|^{2}Φ_{h}|γ′_{M}|η^{2} dx \\ + & c(p, ν, K) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2}} |D(|Du|^{2} − R^{2})_{+}|Φ′_{h}γ_{M}η(η + |Dη|) dx \\ + & c(p, ν, K) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−1}{2}} |D^{2}u|Φ_{h}γ_{M}η(η + |Dη|) dx \\ + & c(p, ν, K) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2}} |D^{2}u|Φ_{h}|γ′_{M}|η(η + |Dη|) dx . \end{matrix} (3.10)

By Young inequality, it follows

\begin{matrix}∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D^{2}u|^{2}Φ_{h}η^{2} dx \\ + ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D(|Du|^{2} − R^{2})_{+}|^{2}Φ′_{h}η^{2} dx \\ ≤ & c(Λ) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−1}{2}} |D^{2}u|^{2}Φ_{h}|γ′_{M}|η^{2} dx \\ + & c ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2}} Φ_{h}γ_{M}(η^{2} + |Dη|^{2}) dx \\ + & c ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2} +1}Φ′_{h}γ_{M}(η^{2} + |Dη|^{2}) dx \\ + & c ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2}} |D^{2}u|Φ_{h}|γ′_{M}|η(η + |Dη|) dx . \end{matrix}  (3.11)

Since Lemma 3.3 implies that (1+|Du|^{2})^{(p−1)/2}|D^{2}u| ∈ L_{loc}^{1}, taking into account that γ′_{M}(x) = 0 if |Du(x)| > 2M, and that (1 + |Du|^{2})^{\frac{1}{2}} |γ′_{M}| ≤ C, with C independent of M, then every term containing the derivative of γ_{M} in (3.11) tends to zero when M → +∞. Setting

V (x) := 1 + R^{2} + (|Du(x)|^{2} − R^{2})_{+},  (3.12)

we get

∫_{B_{r}} V ^{\frac{p−2}{2}} |D^{2}u|^{2}Φ_{h}((|Du|^{2} − R^{2})_{+})η^{2} dx
+ ∫_{B_{r}} V ^{\frac{p−2}{2}} |D(|Du|^{2} − R^{2})_{+}|^{2}Φ′_{h}((|Du|^{2} − R^{2})_{+})η^{2} dx
≤ c ∫_{B_{r}} V ^{\frac{p}{2} +1}Φ′_{h}((|Du|^{2} − R^{2})_{+})(η^{2} + |Dη|^{2}) dx
+ c ∫_{B_{r}} V ^{\frac{p}{2}} Φ_{h}((|Du|^{2} − R^{2})_{+})(η^{2} + |Dη|^{2}) dx .  (3.13)

Passing to the limit as h → +∞, by the monotone convergence theorem it follows that (3.13) holds with Φ and Φ′ in place of Φ_{h} and Φ′_{h}, respectively:

∫_{B_{r}} V ^{\frac{p}{2} +δ−1}(|Du|^{2} − R^{2})_{+}|D^{2}u|^{2}η^{2} dx

+ ∫_{B_{r}} V ^{\frac{p}{2} +δ−2}(1 + R^{2} + (1 + δ)(|Du|^{2} − R^{2})_{+})|D(|Du|^{2} − R^{2})_{+}|^{2}η^{2} dx

≤ c ∫_{B_{r}} V ^{\frac{p}{2} +δ}(1 + R^{2} + (1 + δ)(|Du|^{2} − R^{2})_{+})(η^{2} + |Dη|^{2}) dx
+ c ∫_{B_{r}} V ^{\frac{p}{2} +δ}(|Du|^{2} − R^{2})_{+}(η^{2} + |Dη|^{2}) dx .

Hence (3.3) holds:

\begin{matrix}∫_{B_{r}} V ^{\frac{p}{2} +δ−1}|D(|Du|^{2} − R^{2})_{+}|^{2}η^{2} dx \\ ≤ & c\frac{max\{1, 1 + δ\}}{min\{1, 1 + δ\}} ∫_{B_{r}} V ^{\frac{p}{2} +δ+1}(η^{2} + |Dη|^{2}) dx , \end{matrix}

with c = c(p, ν, K).

Step 2. Let δ > −1 be as in step 1. Define γ = \frac{p}{4} + \frac{δ+1}{2}. Let χ be equal to \frac{n}{n−2} if n > 2, or to any number greater than 1 if n = 2, from the above estimate we have

∫_{B_{r}} |D(V^{γ}η)|^{2} dx ≤ c (1 + γ^{2} \frac{max\{1, 2γ − p/2\}}{min\{1, 2γ − p/2\}})∫_{B_{r}}V^{2γ}(η^{2} + |Dη|^{2}) dx

and by Sobolev inequality we get

[∫_{B_{r}} (V^{2γ}η^{2})^{χ} dx]^{\frac{1}{χ}} ≤ c (1 + γ^{2} \frac{max\{1, 2γ − p/2\}}{min\{1, 2γ − p/2\}}) ∫_{B_{r}} V^{2γ}(η^{2} + |Dη|^{2}) dx . (3.14)

For any i ∈ ℕ set r_{i} = \frac{r}{4} + \frac{r}{4^{i}}. Let η_{i} be a cut-off function in C_{c}^{1}(B_{r_{i}}), such that η_{i} = 1 in B_{r_{i+1}} and η_{i}^{2} + |Dη_{i}|^{2} ≤ c(r)4^{2i}. Given θ > 1, to be chosen later, consider the sequence (γ_{i}) defined by γ_{i+1} = γ_{i}χ, where γ_{1} = \frac{p}{4}θ. Then γ_{i} goes to +∞ when i → +∞. From (3.14) it follows that for all i

[∫_{B_{r_{i+1}}} V^{2γ_{i+1}} dx]^{\frac{1}{χ}} ≤ c4^{2i} (1 + γ_{i}^{2} \frac{max\{1, 2γ_{i} − p/2\}}{min\{1, 2γ_{i} − p/2\}}) ∫_{B_{r_{i}}} V^{2γ_{i}} dx.

Setting c_{i} = [c4^{2i} (1 + γ_{i}^{2} \frac{max\{1,2γ_{i}−p/2\}}{min\{1,2γ_{i}−p/2\}})]^{\frac{1}{2γ_{i}}}, the previous inequality can be rewritten as

||V ||_{L^{2γ_{i+1}} (B_{r_{i+1}} )} ≤ c_{i}||V ||_{L^{2γ_{i}} (B_{r_{i}})} .  (3.15)

Iterating (3.15), we get

||V ||_{L^{2γ_{i+1}} (B_{r_{i+1}} )} ≤ ∏_{j=1}^{i} c_{j}||V ||_{L^{2γ_{1}} (B_{r_{1}} )} .

Since ∏_{j=1}^{i} c_{j} is uniformly bounded in i, passing to the limit with respect to i we get

sup_{B_{r/4}} |V | ≤ C (∫_{B_{r/2}} V ^{\frac{p}{2} θ}(x) dx)^{\frac{2}{pθ}} ,

where C = C(n, p, ν, K, θ, r). On the other hand, since c(R)V (x) ≤ 1 + |Du(x)|^{2} ≤ V (x), we have

sup_{B_{r/4}} |Du| ≤ c [∫_{B_{r/2}}(1 + |Du|^{2})^{\frac{p}{2} θ} dx]^{\frac{1}{pθ}} ,  (3.16)

with c depending also on R. By Theorem 2.5 (i), there exist c̃_{0}, c̃_{1} such that for all (x, ξ) ∈ Ω × ℝ^{nN}

−c̃_{0} + c̃_{1}|ξ|^{p} ≤ f(x, ξ) ≤ K(1 + |ξ|)^{p},

therefore the higher summability property of local minimizers (see [12]), implies that there exists θ_{0} > 1, depending only on n, p, ν, R and K, such that Du is locally in L^{pθ_{0}} and

[− ∫_{B_{r/2}}(1 + |Du|^{2})^{\frac{pθ_{0}}{2}} dx]^{\frac{1}{pθ_{0}}} ≤ c [− ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2}} dx]^{\frac{1}{p}} .  (3.17)

Choosing θ = θ_{0}, (3.16) and (3.17) give (3.2). ■

Suppose now that u is a local minimizer of (3.1), with f satisfying (A1)–(A4) and (H1)–(H4). From Proposition 3.1 u is locally Lipschitz continuous. This fact, together with a suitable iteration argument, allows us to obtain an estimate similar to (3.2), with constants depending only on the constants in (A2)–(A4). Precisely, the following proposition holds:

Proposition 3.5 Let u be a local minimizer of (3.1), where f satisfies (A1)– (A4) and (H1)–(H4), with q < p\frac{n+1}{n}. For any B_{r}(x_{0}) ⊂⊂ Ω, there exist C = C(n, p, q, ν, R, L, r) and β = β(n, p, q) positive constants, such that

sup_{B_{r/4}(x_{0})} |Du|^{p} ≤ C [∫_{B_{r}(x_{0})}(1 + |Du|^{2})^{\frac{p}{2}} dx]^{β} .  (3.18)

Proof. By Proposition 3.1 and Lemma 3.3, u is in W_{loc}^{1,∞}(Ω) ∩ W_{loc}^{2,2}.

Step 1. Given η ∈ C_{c}^{1}(B_{r}), 0 ≤ η ≤ 1, we will show that if if δ > −1 then

\begin{matrix}∫_{B_{r}}(1 + |Du|^{2})^{\frac{p}{2} +δ−1}|D(|Du|^{2} − R^{2})_{+}|^{2}η^{2} dx \\ ≤ & c\frac{max\{1, 1 + δ\}}{min\{1, 1 + δ\}} ∫_{B_{r}}(1 + |Du|^{2})^{q− \frac{p}{2} +δ+1}(η^{2} + |Dη|^{2}) dx , \end{matrix}  (3.19)

with c depending on q, ν and L. The proof of this claim can be obtained following closely step 1 of Proposition 3.1, to which we address for formulas and notations. Since u is locally Lipschitz continuous then the Euler system (3.5) is satisfied with Φ_{h} = Φ and γ_{M} = 1. Using assumption (A3) instead of (H2), (3.9) becomes

|D_{ξ}f(x, ξ)| ≤ c(L)(1 + |ξ|)^{q−1}

and, using (A4) in place of (H3), we obtain (3.10) with p replaced by q in the right-hand side, namely

∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D^{2}u|^{2}Φη^{2}dx,
+ ∫_{B_{r}}(1 + |Du|^{2})^{\frac{p−2}{2}} |D(|Du|^{2} − R^{2})_{+}|^{2}Φ′η^{2} dx
≤ c(q, ν, L) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{q}{2}} |D(|Du|^{2} − R^{2})_{+}|Φ′η(η + |Dη|) dx
+ c(q, ν, L) ∫_{B_{r}}(1 + |Du|^{2})^{\frac{q−1}{2}} |D^{2}u|Φη(η + |Dη|) dx, .

Using the definition of Φ and V, see (3.4) and (3.12), Young inequality yields

∫_{B_{r}} V ^{\frac{p}{2} +δ−1}(|Du|^{2} − R^{2})_{+}|D^{2}u|^{2}η^{2} dx

+ ∫_{B_{r}} V ^{\frac{p}{2} +δ−2}(1 + R^{2} + (1 + δ)(|Du|^{2} − R^{2})_{+})|D(|Du|^{2} − R^{2})_{+}|^{2}η^{2} dx

≤c ∫_{B_{r}} V^{q− \frac{p}{2} +δ}(1 + R^{2} + (1 + δ)(|Du|^{2} − R^{2})_{+})(η^{2} + |Dη|^{2}) dx
+c ∫_{B_{r}} V^{q− \frac{p}{2} +δ}(|Du|^{2} − R^{2})_{+}(η^{2} + |Dη|^{2}) dx

and (3.19) easily follows.

Step 2. Define γ = \frac{p}{4} + \frac{δ+1}{2}. Setting τ = q − p and χ = \frac{n}{n−2} if n > 2, or any number greater than \frac{p}{p−2τ} if n = 2, from (3.19)

∫_{B_{r}}|D(V^{γ}η)|^{2} dx ≤ c(q, ν, L)(1+γ^{2} \frac{max\{1, 2γ − p/2\}}{min\{1, 2γ − p/2\}})∫_{B_{r}}V^{2γ+τ} (η^{2} + |Dη|^{2}) dx

and the Sobolev inequality yields

[∫_{B_{r}} (V^{2γ}η^{2})^{χ} dx]^{\frac{1}{χ}} ≤c (1 + γ^{2} \frac{max\{1, 2γ − p/2\}}{min\{1, 2γ − p/2\}}) ∫_{B_{r}}V^{2γ+τ} (η^{2} + |Dη|^{2}) dx . (3.20)

For every i ∈ ℕ let us define r_{i} = \frac{r}{4} + \frac{r}{4^{i}}. Let η_{i} be a cut-off function in C_{c}^{1}(B_{r_{i}}), such that η_{i} = 1 in B_{r_{i+1}} and η_{i}^{2} + |Dη_{i}|^{2} ≤ c(r)4^{2i}. Let θ > 1 and consider the sequence (γ_{i}) defined as 2γ_{i+1} = 2γ_{i}χ − τ, with γ_{1} = \frac{p}{4}θ. Since \frac{2τ}{p(χ−1)} < 1 < θ then γ_{i} goes to +∞ when i → +∞. From (3.20) it follows that for all i

[∫_{B_{r_{i+1}}} V^{2γ_{i}χ} dx]^{\frac{1}{χ}} ≤ c4^{2i} (1 + γ_{i}^{2} \frac{max\{1, 2γ_{i} − p/2\}}{min\{1, 2γ_{i} − p/2\}}) ∫_{B_{r_{i}}} V^{2γ_{i−1}χ} dx.

Setting c_{i} = [c4^{2i} (1 + γ_{i}^{2} \frac{max\{1,2γ_{i}−p/2\}}{min\{1,2γ_{i}−p/2\}})]^{\frac{1}{2γ_{i}}}, the previous inequality can be rewritten as

||V ||_{L^{2γ_{i}χ}(B_{r_{i+1}} )} ≤ c_{i}||V ||_{L^{2γ_{i−1}χ}(B_{r_{i}})}^{γ_{i−1}χ/γ_{i}} .  (3.21)

Iterating (3.21), we get

||V ||_{L^{2γ_{i}χ}(B_{r_{i+1}} )} ≤ ∏_{j=0}^{i−1} c_{i−j}^{\frac{γ_{i−j}}{γ_{i}} χ^{j}}||V ||_{L^{2γ_{1}+τ} (B_{r_{1}} )}^{\frac{γ_{0}}{γ_{i}} χ^{i}} ,

where γ_{0} = \frac{2γ_{1}+τ}{2χ}. Since ∏_{j=0}^{i−1} c_{i−j}^{\frac{γ_{i−j}}{γ_{i}} χ^{j}} is uniformly bounded, passing to the limit with respect to i we get

sup_{B_{r/4}} |V | ≤ c (∫_{B_{r/2}} V ^{\frac{p}{2} θ+τ} dx)^{μ} ,  (3.22)

where c = c(n, p, q, ν, L, r) and μ = (χ − 1) (pθ \frac{χ−1}{2} − τ)^{−1} .

Step 3. In order to get the result we need to estimate the second term of (3.22). By an iteration technique we shall show that there exist c > 0 and α > 1, depending only on the constants in (A2)–(A4), such that

∫_{B_{r/2}} V ^{\frac{p}{2} θ+τ} dx ≤ c [∫_{B_{r}} V ^{\frac{p}{2}} dx]^{α} .  (3.23)

Fixed ρ < σ < r, let η be a cut-off function such that η = 1 on B_{ρ}, η = 0 outside B_{σ}, |Dη| ≤ 2/(σ − ρ). From (3.20)

[∫_{B_{ρ}} V ^{\frac{p}{2} θχ} dx]^{\frac{1}{χ}} ≤ c\frac{1}{(σ − ρ)^{2}} ∫_{B_{σ}} V ^{\frac{p}{2} θ+τ} dx ,  (3.24)

where c = c(n, p, q, ν, L). Setting s = 1 + \frac{2τ}{pθ}, δ = \frac{χ}{s} and H = V ^{\frac{p}{2} θs}, (3.24) may be rewritten as

[∫_{B_{ρ}} H^{δ} dx]^{\frac{1}{χ}} ≤ c\frac{1}{(σ − ρ)^{2}} ∫_{B_{σ}} H dx .

Let t > χ to be chosen later. By Hölder inequality

∫_{B_{σ}} H dx ≤ [∫_{B_{σ}} H^{δ} dx]^{\frac{1}{t}} [∫_{B_{σ}} H^{\frac{t−δ}{t−1}} dx]^{\frac{t−1}{t}} ,

therefore

[∫_{B_{ρ}} H^{δ} dx]^{\frac{1}{χ}} ≤ c\frac{1}{(σ − ρ)^{2}} [∫_{B_{σ}} H^{δ} dx]^{\frac{1}{t}} [∫_{B_{r}} H^{\frac{t−δ}{t−1}} dx]^{\frac{t−1}{t}} .  (3.25)

For all i ∈ ℕ, consider the last inequality with σ = ρ_{i}, ρ = ρ_{i−1}, where ρ_{i} = r − \frac{r}{2^{i}}. Iterating (3.25) we get

\begin{aligned}∫_{B_{r/2}} H^{δ} & ≤ [∫_{B_{ρ_{i+1}}} H^{δ}]^{(^{\frac{χ}{t}})}^{i} ∏_{j=1}^{i} (c4^{j})^{χ(\frac{χ}{t})^{j−1}} [∫_{B_{r}} H^{\frac{t−δ}{t−1}} ]^{(t−1)(\frac{χ}{t})^{j}} \\ & ≤ [∫_{B_{r}} H^{δ}]^{(^{\frac{χ}{t}})}^{i} ∏_{j=1}^{i} (c4^{j})^{χ(\frac{χ}{t})^{j−1}} [∫_{B_{r}} H^{\frac{t−δ}{t−1}} ]^{(t−1)(\frac{χ}{t})^{j}} . \end{aligned} (3.26)

Since u ∈ W_{loc}^{1,∞}(Ω) the first term on the right hand of (3.26) tends to 1 when i → +∞, while

\begin{aligned}∏_{j=1}^{∞} (c4^{j+1})^{χ(\frac{χ}{t})^{j}} [∫_{B_{r}} H^{\frac{t−δ}{t−1}} ]^{(t−1)(\frac{χ}{t})^{j}} & ≤ c [∫_{B_{r}} H^{\frac{t−δ}{t−1}} ]^{χ \frac{(t−1)}{t−χ}} \\ & = c [∫_{B_{r}} V ^{\frac{p}{2} θs \frac{t−δ}{t−1}} ]^{χ \frac{t−1}{t−χ}} . \end{aligned}

where c = c(n, p, q, t, ν, R, L, r). We choose t in such a way that θ \frac{ps(t−δ)}{2(t−1)} = \frac{p}{2}, that is θs\frac{t−δ}{t−1} = 1. Such a t exists, since by the choice of s, χ and the bound on q (q < p\frac{n+1}{n})

lim_{t→+∞} θs\frac{t − δ}{t − 1} = θs > 1 , θs\frac{χ − δ}{χ − 1} < 1 .

With this choice, since

∫_{B_{r/2}} H dx ≤ [∫_{B_{r/2}} H^{δ} dx]^{\frac{1}{δ}} |B_{r/2}|^{1− \frac{1}{δ}} ,

(3.23) follows with α = s\frac{t−1}{t−χ}. Since c(R)V (x) ≤ 1 + |Du(x)|^{2} ≤ V (x) we finally have (3.18) with β = \frac{pμα}{2}. ■

Remark 3.6 Obvious changes in the proof of Proposition 3.5 imply that estimate (3.18) can be rewritten in such a way that for all ρ < r there exists c_{0} depending on n, p, q, ν, R, L, r, ρ such that

sup_{B_{ρ}(x_{0})} |Du|^{p} ≤ c_{0} (∫_{B_{r}(x_{0})}(1 + |Du|^{p}) dx)^{β} .

4 Approximation and proof of Theorem 1.1

The main tools to prove Theorem 1.1 are two approximation lemmas. In the first lemma we define a sequence of functions (f_{k}) with p-growth, which monotonically converges to f. The second lemma gives a smooth approximation for each f_{k}, with functions satisfying (H1)–(H4).

Lemma 4.1 Let f : Ω × ℝ^{nN} → [0, +∞), f = f(x, ξ), be a Carathéodory function, convex with respect to ξ, which satisfies assumptions (A1)–(A4). Then there exists a sequence (f_{k}) of Carathéodory functions f_{k} : Ω × ℝ^{nN} → [0, +∞), convex with respect to the last variable, monotonically convergent to f, such that

(I) for a.e. x ∈ Ω and every ξ ∈ ℝ^{nN} \ B_{R}(0), f_{k}(x, ξ) = f̃_{k}(x, |ξ|),

(II) for a.e. x ∈ Ω , for every ξ ∈ ℝ^{nN}, and for every k, f_{k}(x, ξ) ≤ f_{k+1}(x, ξ) ≤ f(x, ξ),

(III) f_{k} is p-uniformly convex at infinity with respect to ξ, with \overline{ν} depending only on p and ν,

(IV) for a.e. x ∈ Ω and for every ξ ∈ ℝ^{nN}, there exist L_{1}, independent of k, and \overline{L}_{1}, depending on k, such that

\begin{aligned}f_{k}(x, ξ) & ≤ L_{1}(1 + |ξ|)^{q}, \\ f_{k}(x, ξ) & ≤ \overline{L}_{1}(k)(1 + |ξ|)^{p}, \end{aligned}

(V) using the same notations as in (A4), for every ξ ↦ f_{k}(x, ξ) the vector field x ↦ D_{ξ}^{+}f_{k}(x, ξ) is weakly differentiable and for a.e. x ∈ Ω and for every ξ ∈ ℝ^{nN} \ B_{R}(0)

\begin{aligned}|D_{x}D_{ξ}^{+}f_{k}(x, ξ)| & ≤ L_{1}(1 + |ξ|)^{q−1}, \\ |D_{x}D_{ξ}^{+}f(x, ξ)| & ≤ \overline{L}_{1}(k)(1 + |ξ|)^{p−1}. \end{aligned}

Proof. From Theorem 2.5 (ii) there exist c_{0} = c_{0}(p, q, ν, R, L) and c_{1} = c_{1}(p, ν) positive constants and a function g : Ω × ℝ^{nN} → [−c_{0}, +∞) such that

f(x, ξ) = c_{1}(1 + |ξ|^{2})^{\frac{p}{2}} + g(x, ξ),

with g convex at infinity, and there exists g̃ : Ω × [R, +∞) → [−c_{0}, +∞) such that g̃(x, |ξ|) = g(x, ξ) for any ξ ∈ ℝ^{nN} \ B_{R}(0). Since nN > 1, for a.e. x ∈ Ω, t ↦ g̃(x, t) is convex and increasing. For any k ∈ ℕ, k > R, using the same notations as in (A4), define g̃_{k} : Ω × [R, +∞) → [−c_{0}, +∞) as follows

\begin{matrix}g̃_{k}(x, t) = g̃(x, t) & ∀(x, t) ∈ Ω × [R, k] \\ g̃_{k}(x, t) = g̃(x, k) + D_{t}^{+}g̃(x, k)(t − k) & ∀(x, t) ∈ Ω × (k, +∞). \end{matrix}

For a.e. x ∈ Ω, t ↦ g̃_{k}(x, t) is convex and increasing in [R, +∞) and g̃_{k}(x, t) ≤ g̃_{k+1}(x, t) ≤ g̃(x, t). (A3) and the definition of g̃_{k}(x, t) give

\begin{aligned}g̃_{k}(x, t) & ≤ L(1 + t)^{q}, \\ g̃_{k}(x, t) & ≤ c(q, L, k)(1 + t)^{p}. \end{aligned}

If t ∈ (k, +∞) D_{t}g̃_{k}(x, t) = D_{t}^{+}g̃(x, k) and (A4) implies that |D_{x}D_{t}^{+}g̃(x, k)| ≤ L(1 + k)^{q−1}, therefore for a.e. x and every t > R

\begin{aligned}|D_{x}D_{t}^{+}g̃(x, t)| & ≤ c(1 + t)^{q−1} \\ |D_{x}D_{t}^{+}g̃(x, t)| & ≤ c(k)(1 + t)^{p−1}. \end{aligned}

For any k, define g_{k} : Ω × ℝ^{nN} → [−c_{0}, +∞), g_{k}(x, ξ) = g(x, ξ) if |ξ| < R and g_{k}(x, ξ) = g̃_{k}(x, |ξ|) otherwise. Setting f_{k} : Ω × ℝ^{nN} → [0, +∞)

f_{k}(x, ξ) := c_{1}(1 + |ξ|^{2})^{\frac{p}{2}} + g_{k}(x, ξ),

the result easily follows. ■

Lemma 4.2 Let f_{k} be as in Lemma 4.1. Fixed an open set A ⊂⊂ Ω, there exists a sequence of C^{2}-functions f_{kh} : A × ℝ^{nN} → [0, +∞) with the following property. For any compact set K ∈ ℝ^{nN} and for any δ > 0 there exists A_{δ} ⊂ A with |A_{δ}| < δ such that

lim_{h→∞} f_{kh} = f_{k}  uniformly on (A \ A_{δ}) × K.  (4.1)

Moreover f_{kh} satisfies (A1)–(A4) and (H1)–(H4). More precisely,

(A1) for every x ∈ A and ξ ∈ ℝ^{nN} \ B_{R+1}(0), f_{kh}(x, ξ) = f̃_{kh}(x, |ξ|),

(A2) there exists ν̃ depending on p, ν and R, but neither on k nor h, such that for every x ∈ A and for every ξ_{1}, ξ_{2} ∈ ℝ^{nN} \ B_{R+1}(0) endpoints of a segment contained in the complement of B_{R+1}(0),

\frac{1}{2}[f_{kh}(x, ξ_{1})+f_{kh}(x, ξ_{2})] ≥ f_{kh}(x, \frac{ξ_{1} + ξ_{2}}{2})+ν̃(1+|ξ_{1}|^{2}+|ξ_{2}|^{2})^{\frac{p−2}{2}} |ξ_{1}−ξ_{2}|^{2},

(A3) there exists L_{2}, depending neither on k nor on h, such that for all (x, ξ) ∈

A × ℝ^{nN}

f_{kh}(x, ξ) ≤ L_{2}(1 + |ξ|)^{q},

(A4) for every x ∈ A and ξ ∈ ℝ^{nN} \ B_{R+1}(0)

|D_{ξx}f_{kh}(x, ξ)| ≤ L_{2}(1 + |ξ|)^{q−1},

(H1) f_{kh} is in C^{2}(A × ℝ^{nN}),

(H2) there exists \overline{L}_{2}(k), independent of h, such that for all (x, ξ) ∈ A × ℝ^{nN}

f_{kh}(x, ξ) ≤ \overline{L}_{2}(k)(1 + |ξ|)^{p},

(H3) there exist ϵ_{h} > 0, independent of k, and Λ_{1}(h) such that for all (x, ξ) ∈ A × ℝ^{nN} and for all λ ∈ ℝ^{nN}

ϵ_{h}(1 + |ξ|^{2})^{\frac{p−2}{2}} |λ|^{2} ≤ ⟨D_{ξξ}f_{kh}(x, ξ)λ, λ⟩ ≤ Λ_{1}(h)(1 + |ξ|^{2})^{\frac{p−2}{2}} ,

(H4) there exists \overline{Λ}_{1}(k), independent of h, such that for every x ∈ A and ξ ∈ ℝ^{nN} \ B_{R+1}(0),

|D_{ξx}f_{kh}(x, ξ)| ≤ \overline{Λ}_{1}(k)(1 + |ξ|^{2})^{\frac{p−1}{2}} .

Proof. Let f_{k} be as in Lemma 4.1, that is

f_{k}(x, ξ) = c_{1}(1 + |ξ|^{2})^{\frac{p}{2}} + g_{k}(x, ξ).  (4.2)

Fixed an open set A ⊂⊂ Ω and an infinitesimal positive sequence ε_{h}, with h large enough, consider B_{1}^{n} ⊂ ℝ^{n} and B_{1}^{nN} ⊂ ℝ^{nN} the unit balls centered in 0. Let σ : B_{1}^{n} → [0, +∞) and ρ : B_{1}^{nN} → [0, +∞) be two radially symmetric mollifiers such that ∫_{B_{1}^{n}} σ = 1 and ∫_{B_{1}^{nN}} ρ = 1. Define F_{kh} : A×ℝ^{nN} → [0, +∞),

F_{kh}(x, ξ):=∫_{B_{1}^{n}}∫_{B_{1}^{nN}} σ(y)ρ(η) f_{k}(x + ε_{h}y, ξ + ε_{h}η) dη dy .

Since g_{k}(x, ξ) = g̃_{k}(x, |ξ|) if |ξ| ≥ R, then there exists F̃_{kh} : A × [R + 1, +∞) → [0, +∞) such that F_{kh}(x, ξ) = F̃_{kh}(x, |ξ|) for every x ∈ A and ξ ∈ ℝ^{nN} \B_{R+1}(0). Moreover, F_{kh} is convex with respect to ξ in ℝ^{nN}. Fixed a compact subset K in R^{nN}, for any m ∈ ℕ there exists a finite number of open balls B_{\frac{1}{m}} (ξ_{j}^{m}) such that K ⊂ ∪_{j=1}^{i(m)}B_{\frac{1}{m}} (ξ_{j}^{m}). For every m and j = 1, …, i(m), x ↦ ∫_{B_{1}^{n}}σ(y)f_{k}(x + ε_{h}y, ξ_{j}^{m}) dy converges in L^{1} to x ↦ f_{k}(x, ξ_{j}^{m}), as h goes to +∞. By a diagonal argument, eventually passing to subsequences, we have that for every m ∈ ℕ, j = 1, …, i(m) and for a.e. x ∈ A

lim_{h→∞} ∫_{B_{1}^{n}}σ(y)[f_{k}(x + ε_{h}y, ξ_{j}^{m}) − f_{k}(x, ξ_{j}^{m})] dy = 0

By Egorov Theorem, for any δ > 0 there exists A_{δ} ⊂ A with |A_{δ}| < δ such that for any m

lim_{h→∞} sup_{x∈A\A_{δ}} ∫_{B_{1}^{n}}σ(y)[f_{k}(x + ε_{h}y, ξ_{j}^{m}) − f_{k}(x, ξ_{j}^{m})] dy = 0 ,  (4.3)

for every j = 1, …, i(m). We claim that F_{kh} converges uniformly to f_{k} in (A \ A_{δ}) × K as h → +∞, that is

lim_{h→∞} sup_{(x,ξ)∈(A\A_{δ})×K} |F_{kh}(x, ξ) − f_{k}(x, ξ)| = 0.  (4.4)

In fact, given ξ ∈ K, for every m there exists j = 1, …, i(m) such that ξ ∈ B_{\frac{1}{m}} (ξ_{j}^{m}). Consider

F_{kh}(x, ξ) − f_{k}(x, ξ)

= ∫_{B_{1}^{n}}∫_{B_{1}^{nN}} σ(y)ρ(η)[f_{k}(x + ε_{h}y, ξ + ε_{h}η) − f_{k}(x + ε_{h}y, ξ)]dη dy

+ ∫_{B_{1}^{n}}σ(y)[f_{k}(x + ε_{h}y, ξ) − f_{k}(x + ε_{h}y, ξ_{j}^{m})] dy
+ ∫_{B_{1}^{n}}σ(y)[f_{k}(x + ε_{h}y, ξ_{j}^{m}) − f_{k}(x, ξ_{j}^{m})] dy
+ (f_{k}(x, ξ_{j}^{m}) − f_{k}(x, ξ))
= I_{1}^{h,m}(x, ξ) + I_{2}^{h,m}(x, ξ) + I_{3}^{h,m}(x, ξ) + I_{4}^{h,m}(x, ξ).  (4.5)

The first inequality in (IV) of the previous lemma and the convexity of f_{k} imply that there exists c > 0 such that for a.e. x ∈ Ω and for every λ, η ∈ ℝ^{nN}

|f_{k}(x, λ) − f_{k}(x, η)| ≤ c(1 + |λ| + |η|)^{q−1}|λ − η|;

therefore

|I_{1}^{h,m}(x, ξ) + I_{2}^{h,m}(x, ξ) + I_{4}^{h,m}(x, ξ)| ≤ c (ε_{h} + \frac{1}{m}) sup_{ξ∈K}(1 + |ξ|)^{q−1}.

This estimate, together with (4.3), implies (4.4). Define

f_{kh}(x, ξ) := F_{kh}(x, ξ) + ε_{h}(1 + |ξ|^{2})^{\frac{p}{2}} .

For each h, f_{kh} ∈ C^{2} and f_{kh} is radial with respect to ξ if |ξ| ≥ R + 1. It is easy to check that Theorem 2.5 gives (A2) and that property (IV) implies (A3) and (H2). (A4) and (H4) are an easy consequence of (V). The ellipticity condition in (H3) follows by the convexity of F_{kh}(x, ξ), while the right inequality in (H3) is a consequence of the definition of g̃_{k}. ■

Now we are ready to prove the main result.

Proof of Theorem 1.1. Let u be a local minimizer of F in (1.1). Fixed A ⊂⊂ Ω, let f_{k}, f_{kh} be defined as in Lemma 4.1 and 4.2. From (A2) and Theorem 2.5 (i), there exists c_{1} > 0 such that

|ξ|^{p} ≤ c_{1}(1 + f_{kh}(x, ξ)), ∀k, h.  (4.6)

Fixed B_{r}(x_{0}) in A, let v_{kh} be the solution of the problem

min {F_{kh}(w; B_{r}(x_{0})) := ∫_{B_{r}(x_{0})} f_{kh}(x, Dw) dx : w ∈ u + W_{0}^{1,p}(B_{r}(x_{0}))} .

From (4.6), the minimality of v_{kh} implies

∫_{B_{r}(x_{0})} |Dv_{kh}|^{p} dx ≤ c_{1} ∫_{B_{r}(x_{0})}(1 + f_{kh}(x, Du)) dx.  (4.7)

By (H2), the dominated convergence theorem, recalling that f_{k}(x, ξ) ≤ f(x, ξ), we have

lim_{h} ∫_{B_{r}(x_{0})} f_{kh}(x, Du) dx = ∫_{B_{r}(x_{0})} f_{k}(x, Du) dx ≤ ∫_{B_{r}(x_{0})} f(x, Du) dx. (4.8)

From (4.7) and (4.8)

lim inf_{h} ∫_{B_{r}(x_{0})} |Dv_{kh}|^{p} dx ≤ c_{1} ∫_{B_{r}(x_{0})}(1 + f(x, Du)) dx.  (4.9)

Therefore, up to subsequences, v_{kh} weakly converges in W^{1,p} to a function v_{k} ∈ u + W_{0}^{1,p}(B_{r}(x_{0})). For any k and h, F_{kh} satisfies the assumptions of Proposition 3.5 with R+1 in place of R. Remark 3.6, the minimality of v_{kh} and (4.8) imply that for any ρ < r there exists β = β(n, p, q) and c_{2}, independent of h and k, such that

lim inf_{h} sup_{B_{ρ}(x_{0})} |Dv_{kh}|^{p} ≤ c_{2} lim inf_{h} [∫_{B_{r}(x_{0})}(1 + f_{kh}(x, Dv_{kh})) dx]^{β}

≤ c_{2} [∫_{B_{r}(x_{0})}(1 + f(x, Du)) dx]^{β} =: M.

(4.10)

Up to a subsequence, v_{kh} converges to v_{k} w^{∗} − W^{1,∞}(B_{ρ}(x_{0})) for any ρ < r and

∫_{B_{r}(x_{0})} |Dv_{k}|^{p} dx ≤ c_{1} ∫_{B_{r}(x_{0})}(1 + f(x, Du)) dx,  (4.11)

sup_{B_{ρ}(x_{0})} |Dv_{k}|^{p} ≤ c_{2} [∫_{B_{r}(x_{0})}(1 + f(x, Du)) dx]^{β} .  (4.12)

Therefore, up to subsequences, v_{k} weakly converges in W^{1,p} and weakly^{∗} in W_{loc}^{1,∞}(B_{r}(x_{0})) to a function v ∈ u + W_{0}^{1,p}(B_{r}(x_{0})). Let us show that v is a minimizer of F(w; B_{r}(x_{0})). Fix ρ < r and k_{0} ∈ ℕ. Since if k > k_{0}, f_{k}(x, ξ) ≥ f_{k_{0}}(x, ξ) for any x and ξ, by the lower semicontinuity of w ↦ ∫_{B_{r}} f_{k_{0}}(x, Dw) dx,

∫_{B_{ρ}(x_{0})} f_{k_{0}}(x, Dv_{k}) dx ≤ lim inf_{h} ∫_{B_{ρ}(x_{0})} f_{k_{0}}(x, Dv_{kh}) dx

≤ lim inf_{h} ∫_{B_{ρ}(x_{0})} f_{k}(x, Dv_{kh}) dx

By Lemma 4.2, fixed K = {ξ ∈ ℝ^{nN} : |ξ| ≤ M + 1}, for every δ > 0 there exists A_{δ} with |A_{δ}| < δ such that f_{kh} converges to f_{k} uniformly in (A\A_{δ})×K. Thus

lim sup_{h} ∫_{B_{ρ}(x_{0})\A_{δ}} f_{k}(x, Dv_{kh}) dx = lim sup_{h} ∫_{B_{ρ}(x_{0})\A_{δ}} f_{kh}(x, Dv_{kh}) dx

and, from (IV) and (4.10), there exists c independent of δ, such that

lim sup_{h} ∫_{B_{ρ}(x_{0})∩A_{δ}} f_{k}(x, Dv_{kh}) dx ≤ cδ\overline{L}_{1}(k)(1 + M) .

Therefore

lim inf_{h}∫_{B_{ρ}(x_{0})}f_{k}(x, Dv_{kh}) dx ≤ lim sup_{h}∫_{B_{r}(x_{0})}f_{kh}(x, Dv_{kh}) dx+cδ\overline{L}_{1}(k)(1 + M).

Letting δ go to 0, from the previous estimates, the minimality of v_{kh} and (4.8) we have

∫_{B_{ρ}(x_{0})} f_{k_{0}}(x, Dv_{k}) dx ≤ ∫_{B_{r}(x_{0})} f(x, Du) dx

and, by the lower semicontinuity,

∫_{B_{ρ}(x_{0})} f_{k_{0}}(x, Dv) dx ≤ lim inf_{k} ∫_{B_{ρ}(x_{0})} f_{k_{0}}(x, Dv_{k}) dx ≤ ∫_{B_{r}(x_{0})} f(x, Du) dx.

Letting k_{0} → ∞ and ρ → r we finally have

∫_{B_{r}(x_{0})} f(x, Dv) dx ≤ ∫_{B_{r}(x_{0})} f(x, Du) dx.

Finally, passing to the limit in (4.12) with respect to k

sup_{B_{r/4}(x_{0})} |Dv|^{p} ≤ c [∫_{B_{r}(x_{0})}(1 + f(x, Du)) dx]^{β} ,

and from Lemma 2.6 there exists R_{0} = R_{0}(p, q, ν, R, L) such that

sup_{B_{r/4}(x_{0})} |Du| ≤ sup_{B_{r/4}(x_{0})} |Dv| + sup_{B_{r/4}(x_{0})} |Du + Dv| ≤ 3 sup_{B_{r/4}(x_{0})} |Dv| + 2R_{0}

and the thesis follows. ■

References

[1] E. Acerbi, N. Fusco, Regularity of minimizers of non-quadratic functionals: the case 1 < p < 2, J. Math. Anal. Appl. 140 (1989) 115–135.

[2] M. Bildhauer, M. Fuchs, G. Mingione, A priori gradient bounds and local C^{1,α}-estimates for (double) obstacle problems under non-standard growth conditions, Z. Anal. Anwendungen 20 (2001) 959–985.

[3] A. Coscia, G. Mingione, Hölder continuity of the gradient of p(x)-harmonic mappings C.R. Acad. Sci. Paris 328 (1999) 363–368.

[4] B. Dacorogna, Direct methods in the Calculus of Variations. Appl. Math. Sci. 78, Springer-Verlag, Heidelberg and New York, 1989.

[5] L. Esposito, F. Leonetti, G. Mingione, Regularity results for minimizers of irregular integrals with (p, q) growth, Forum Mathematicum 14 (2002) 245–272.

[6] L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with (p, q) growth, Preprint 266 (2001) Dip. Mat. Univ. Parma.

[7] I. Fonseca, N. Fusco, Regularity results for anisotropic image segmentation models, Ann. Scuola Norm. Sup. Pisa 24 (1997) 463–499.

[8] I. Fonseca, N. Fusco, P. Marcellini An existence result for a nonconvex variational problem via regularity, ESAIM: Control, Optimization and Calculus of Variations, to appear.

[9] N. Fusco, G. Mingione, C. Trombetti, Regularity of minimizers for a class of anisotropic free discontinuity problems, J. Convex Anal. 8 (2001).

[10] M. Giaquinta, Multiple integrals in calculus of variations and nonlinear elliptic systems, Ann. Math. Stud. 105, Princeton Univ. Press, 1983.

[11] M. Giaquinta, Growth conditions and regularity: a counterexample, Manuscripta Math. 59 (1987) 245–248.

[12] M. Giaquinta, G. Modica, Regularity results for some classes of higher order non linear elliptic systems, J. Reine Angew. Math. 311-312 (1979) 145–169.

[13] M. Giaquinta, G. Modica, Remarks on the regularity of the minimizers of certain degenerate functionals, Manuscripta Math. 57 (1986) 55–99.

[14] E. Giusti, Metodi diretti nel calcolo delle variazioni, U.M.I., Bologna, 1994.

[15] S.M. Kozlov, Geometric aspects of averaging, (Russian) Uspekhi Mat. Nauk 44 (1989) 79–120. (English transl.: Russian Math. Surveys 44 (1989) 91–144).

[16] P. Marcellini, Un exemple de solution discontinue d’un problème dans le cas scalaire, Preprint 8 Ist. Mat. Univ. Firenze (1987).

[17] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Rat. Mech. Anal. 105 (1989) 267–284.

[18] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations 90 (1991) 1–30.

[19] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993) 296–333.

[20] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa 23 (1996) 1–25.

[21] E. Mascolo, A.P. Migliorini, Everywhere regularity for vectorial functionals with general growth, Preprint (2001) Dip. di Mat. Univ. di Firenze.

[22] M. Ruzicka, Electrorheological fluids: Modeling and mathematical theory, Lecture Notes in Mathematics, 1748. Springer-Verlag, Berlin, 2000.

[23] K. Uhlenbeck, Regularity for a class of nonlinear elliptic systems, Acta Math. 138 (1977) 219–240.

[24] V.V. Zhikov, Lavrentiev phenomenon and homogenization for some variational problems, C. R. Acad. Sci. Paris Ser. I Math. 316 (1993) 435–439.