Hölder Continuity for Local Minimizers of a Nonconvex Variational Problem

Giovanni Cupini and Anna Paola Migliorini

Dipartimento di Matematica “U. Dini”, Università di Firenze Viale Morgagni 67/A, 50134 - Firenze, Italy.

Abstract

We consider integral functionals of the Calculus of Variations where the energy density is a continuous function with p-growth, p > 1, uniformly convex at infinity with respect to the gradient variable. We prove that local minimizers are α-Hölder continuous for all α < 1.

Key words: Local minimizer, regularity, nonconvex functional.

AMS classification: 49N60.

1 Introduction

Consider the integral functional of the Calculus of Variations

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

where Ω is an open bounded subset of ℝ^{N} and f = f(x, u, ξ) : Ω × ℝ × ℝ^{N} → [0, +∞) is a Carathéodory function, i.e. f is measurable in x and continuous in (u, ξ), satisfying the p-growth condition (p > 1)

|ξ|^{p} ≤ f(x, u, ξ) ≤ L(1 + |ξ|^{p}).

A function u ∈ W_{loc}^{1,p}(Ω) is a local minimizer of F in Ω if

F (u; spt (v − u)) ≤ F (v; spt (v − u)) ,

for every v ∈ W_{loc}^{1,p}(Ω) such that spt (v − u) ⊂⊂ Ω. Well known results due to Giaquinta and Giusti [12, 14] ensures that local minimizers of F are locally α-Hölder continuous for some α < 1. According to Meyers’ example in [18], when f is not continuous in Ω × ℝ × ℝ^{N}, the α-Hölder continuity for all α < 1 cannot be achieved, even if f is twice differentiable and uniformly convex with respect to ξ. However, if f is of class C^{2} in ξ, uniformly elliptic in this variable and for every x, y ∈ Ω, u, v ∈ ℝ and ξ ∈ ℝ^{N}

|f(x, u, ξ) − f(y, v, ξ)| ≤ ω(|x − y| + |u − v|)(1 + |ξ|^{p}) ,  (1.2)

where ω(t) is a modulus of continuity that grows as t^{δ}, δ > 0, then Du is Hölder continuous (see Giaquinta-Giusti [13], Giaquinta-Modica [15], Manfredi [17]). Recent results show that failing the differentiability of the integrand, the Lipschitz continuity of local minimizers, or at least the Hölder continuity for any exponent, still holds under suitable convexity assumptions. Fonseca and Fusco in [8] study the regularity of local minimizers of functionals with a nondifferentiable integrand f = f(ξ) independent of (x, u). They prove that if f has p-growth and satisfies a uniform convexity condition, i.e. there exists a constant ν > 0 such that for all ξ, η in ℝ^{N}

f (\frac{ξ + η}{2} ) ≤ \frac{1}{2}f(ξ) + \frac{1}{2}f(η) − ν(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} ,  (1.3)

then local minimizers are locally Lipschitz continuous. Notice that when f is of class C^{2}, then (1.3) is equivalent to the usual uniform ellipticity condition

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

for some ν_{0} > 0. The general case f = f(x, u, ξ) has been studied by Cupini, Fusco and Petti in [2]. They prove that if (1.2) and (1.3) hold, then local minimizers are locally α-Hölder continuous for all α < 1. We remark that even in the simple case f = f(x, ξ), N = p = 2, the Lipschitz continuity cannot be achieved (see Example 3.2 in [2]). Related results can be found in [3, 6, 10]. In a recent paper Fonseca, Fusco and Marcellini [9] establish existence and regularity of minimizers of energy integrals with f = f(x, ξ) subject to Dirichlet boundary conditions, where the uniform convexity property of f is satisfied at infinity, i.e. (1.3) holds for all x ∈ Ω and every ξ, η endpoints of a segment contained in the complement of a ball B_{R}(0) in ℝ^{N}. In particular, they prove the Lipschitz continuity of local minimizers when f is a continuous function with p-growth, uniformly convex at infinity, such that for |ξ| > R the vector field x ↦ D_{ξ}f(x, ξ) is weakly differentiable with |D_{ξx}f(x, ξ)| ≤ L(1 + |ξ|^{p−1}). We explicitly notice that in the particular case f = f(ξ) the differentiability assumption can be removed. In this case the p-growth and the uniform convexity at infinity are sufficient conditions to the Lipschitz continuity of local minimizes (see Theorem 2.2 below).

In this paper we consider the general case of nonconvex integrands f = f(x, u, ξ) with p-growth, both uniformly convex and continuous at infinity and we prove that local minimizers of F are α-Hölder continuous for any exponent α < 1. More precisely, we prove the following Theorem 1.1 Let f : Ω × ℝ × ℝ^{N} → [0, +∞) be a Carathéodory function satisfying

(A1) there exist p > 1 and L > 0 such that for a.e. x ∈ Ω, for every u ∈ ℝ and ξ ∈ ℝ^{N}

0 ≤ f(x, u, ξ) ≤ L(1 + |ξ|^{p}) ;

(A2) f is uniformly convex at infinity with respect to ξ, i.e. there exist R > 0 and ν > 0 such that if the segment [ξ, η] is contained in the complement of the ball B_{R}(0), then for a.e. x ∈ Ω and for every u ∈ ℝ

f(x, u, \frac{ξ + η}{2})≤ \frac{1}{2}f(x, u, ξ) + \frac{1}{2}f(x, u, η) − ν(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} ;

(A3) f is continuous at infinity with respect to the pair (x, u), i.e. for every x, y ∈ Ω, for every u, v ∈ ℝ and ξ ∈ ℝ^{N} \ B_{R}(0)

|f(x, u, ξ) − f(y, v, ξ)| ≤ ω(|x − y| + |u − v|)(1 + |ξ|^{p}) ,

where ω : [0, +∞) → [0, +∞) is a continuous, increasing, bounded function such that ω(0) = 0.

If u ∈ W_{loc}^{1,p}(Ω) is a local minimizer of the functional F defined in (1.1), then u ∈ C_{loc}^{0,α}(Ω) for all α < 1.

The plan of the paper is the following. In Section 2 we collect preliminary results concerning functions satisfying (A1) and (A2), higher integrability results for minimizers of functionals with p-growth and an iteration lemma. In Section 3 we study the regularity of a local minimizer u of

I(u; Ω) =: ∫_{Ω} f(x, Du(x)) dx,

when f = f(x, ξ) is a convex function with respect to ξ and satisfies (A1)–(A3). We approximate I(u; Ω) with a sequence of functionals (I_{h}) with integrands f_{h} = f_{h}(x, ξ) of class C^{1} with respect to ξ, satisfying the same assumptions as f. Fixed B_{r}(x_{0}) ⊂⊂ Ω, let u_{h} be the minimizer of I_{h}(w; B_{r}(x_{0})) in the Dirichlet class u+W_{0}^{1,p}(B_{r}(x_{0})), where u is the local minimizer of the functional I. For each u_{h} we establish an integral estimate with constants independent of h. Moreover, we prove that u_{h} converges in the weak topology of W^{1,p} to a function u_{∞}, which turns out to be a minimizer of I in u + W_{0}^{1,p}(B_{r}(x_{0})). Passing to the limit in the integral estimate proved for each u_{h}, we infer that the estimate is satisfied by u_{∞} and, from the uniform convexity at infinity, by u too. From this estimate the Hölder continuity of u for any exponent follows by a classical argument.

In Section 4 we prove Theorem 1.1. Firstly we consider the supplementary assumption of convexity of f with respect to ξ. We compare the local minimizer u of F with the minimizer of a functional I of the type studied in Section 3. An essential tool for this comparison is the Ekeland variational principle. The convexity assumption on f is removed via a relaxation argument.

2 Preliminary results

Let us consider the integral functional

F(u; Ω) := ∫_{Ω} f(x, u(x), Du(x)) dx ,  (2.1)

where Ω is a bounded open subset of ℝ^{N} and u : Ω → ℝ is a scalar function in W_{loc}^{1,p}(Ω), with p > 1. We denote by B_{r}(x_{0}) the open ball with center x_{0} and radius r; we omit x_{0} if no confusion may arise. In the sequel, c is a positive constant which may take different values from line to line. We recalled the definition of local minimizer of F in the Introduction. More generally, u is a Q-minimizer of F if there exists Q ≥ 1 such that

F (u; spt (v − u)) ≤ QF (v; spt (v − u)) ,

for all v ∈ W_{loc}^{1,p}(Ω) such that spt (v − u) ⊂⊂ Ω. We state some consequences of assumptions (A1) and (A2), proved in [9] in the case f = f(ξ). The generalization to our case is straightforward.

Theorem 2.1 Let f = f(x, u, ξ) : Ω × ℝ × ℝ^{N} → [0, +∞), be a Carathéodory function. Then (A1) and (A2) imply:

(i) there exist c_{1}(ν) and c_{2}(p, L, R, ν) such that for a.e. x ∈ Ω and for every (u, ξ) ∈ ℝ × ℝ^{N}

f(x, u, ξ) ≥ c_{1}|ξ|^{p} − c_{2} ;

(ii) there exist R_{0} > R and ν_{0} > 0 depending only on p, L, R and ν, such that for a.e. x ∈ Ω and for every u ∈ ℝ and ξ ∈ ℝ^{N} \ B_{R_{0}}(0) there exists q_{ξ}(x, u) ∈ ℝ^{N} such that

|q_{ξ}(x, u)| ≤ c(p, L)(1 + |ξ|)^{p−1},

and for every η ∈ ℝ^{N}

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

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

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

We underline that Theorem 2.7 of [9] entails the following regularity result.

Theorem 2.2 Let f = f(ξ) : ℝ^{N} → [0, +∞) be a continuous function with p-growth, satisfying the uniform convexity (1.3) at infinity. If u ∈ W_{loc}^{1,p}(Ω) is a local minimizer of the functional

∫_{Ω} f(Du(x)) dx ,

then u is locally Lipschitz continuous in Ω. Moreover, there exists c, depending on N, p, L, R and ν, such that for every B_{r}(x_{0}) ⊂⊂ Ω

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

Now we state a simple algebraic result that will be useful in the sequel (see [16], Lemma 8.5 or [2], Lemma 2.2).

Lemma 2.3 If p > 1, there exists c > 0 such that for every ξ, η ∈ ℝ^{N}

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

We will use some regularity results of Q-minimizers of integral functionals. The following result is contained in Theorem 3.1 of [14].

Lemma 2.4 Let B_{r}(x_{0}) ⊂⊂ Ω and ϕ ∈ L^{p}(B_{r}(x_{0})). If u ∈ W_{loc}^{1,p}(B_{r}(x_{0})) is a Q-minimizer of the functional

w ↦ ∫_{B_{r}} (1 + |Dw(x)|^{p} + |ϕ(x)|^{\frac{p+1}{2}} ) dx ,  (2.2)

then there exists τ > 1 such that u ∈ W_{loc}^{1,pτ} (B_{r}). Moreover, there exists c = c(N, p, Q) such that for every B_{ρ}(x_{1}) ⊂ B_{r}(x_{0})

(− ∫_{B_{ρ/2}(x_{1})} |Du|^{pτ} dx)^{\frac{1}{τ}} ≤ c − ∫_{B_{ρ}(x_{1})}(1 + |Du|^{p} + |ϕ|^{p}) dx .

We need also an up-to-the-boundary higher integrability result (see e.g. Theorem 6.8 of [16] and Lemma 2.7 in [2]).

Lemma 2.5 Let h : B_{2r}(x_{0}) × ℝ^{N} → ℝ be a Carathéodory function such that |ξ|^{p} ≤ h(x, ξ) ≤ L(1 + |ξ|^{p}). If u_{0} ∈ W^{1,q}(B_{2r}(x_{0})), for a certain q > p, and v is a minimizer of the functional

w ↦ ∫_{B_{r}(x_{0})} h(x, Dw(x)) dx

in the Dirichlet class u_{0} + W^{1,p}(B_{r}(x_{0})), then there exist s ∈ (p, q) and c > 0, depending on N, p and L, but neither on u_{0} nor r, such that v ∈ W^{1,s}(B_{r}(x_{0})) and

[− ∫_{B_{r}(x_{0})} |Dv|^{s} dx]^{\frac{1}{s}} ≤ c [− ∫_{B_{2r}(x_{0})}(1 + |Du_{0}|^{q}) dx]^{\frac{1}{q}} .

Finally, we state an iteration lemma (see the proof of Proposition 3.7 in [7]).

Lemma 2.6 Let Z, ψ, ω : [0, T] → [0, +∞) be bounded and increasing functions, ω continuous and ω(0) = 0. Suppose that there exist α, β, γ, c_{1} > 0 such that for every ϵ > 0 there exist c_{2}(ϵ) > 0 such that

Z(t) ≤ c_{1} [(\frac{t}{s})^{α} + ω(s) + ϵ] Z(s) + c_{2}(ϵ)s^{γ}ψ(s)

for any 0 < t < s ≤ T. Then for every 0 < δ < α, there exists T_{0} < T such that

Z(t) ≤ c_{3} (\frac{t}{s})^{α−δ} Z(s) + c_{4}t^{γ}ψ(s),

whenever 0 < t < s ≤ T_{0}. Here T_{0} and c_{3} are positive constants depending only on α, β, δ and c_{1}, while c_{4} depends also on γ.

3 Regularity in the case f = f(x, Du)

In this section we consider the functional

I(u; Ω) := ∫_{Ω} f(x, Du(x)) dx ,  (3.1)

where f = f(x, ξ) : Ω × ℝ^{N} → [0, +∞) is a Carathéodory function, satisfying the assumptions (A1)–(A3), that in this case can be stated as follows:

(A1) the p-growth condition, i.e. for a.e. x ∈ Ω and for every ξ ∈ ℝ^{N}

0 ≤ f(x, ξ) ≤ L(1 + |ξ|^{p});

(A2) the uniform convexity at infinity, i.e. there exist R and ν > 0 such that for a.e. x ∈ Ω if [ξ, η] ⊂ ℝ^{N} \ B_{R}(0), then

f (x, \frac{ξ + η}{2} ) ≤ \frac{1}{2}f(x, ξ) + \frac{1}{2}f(x, η) − ν(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} ;

(A3) the uniform continuity in x at infinity, i.e. for every x, y ∈ Ω and for every ξ ∈ ℝ^{N} \ B_{R}(0)

|f(x, ξ) − f(y, ξ)| ≤ ω(|x − y|)(1 + |ξ|^{p}) ,

where ω : [0, +∞) → [0, +∞) is a continuous, increasing and bounded function such that ω(0) = 0. Without loss of generality, we can assume ω to be concave.

Recall that by Theorem 2.1 (ii) there exists R_{0} > R and ν_{0} > 0 such that

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},  (3.2)

for a.e. x ∈ Ω and for every ξ ∈ ℝ^{N} \ B_{R_{0}}(0) and η ∈ ℝ^{N}, where q_{ξ}(x) ∈ ℝ^{N} satisfies

|q_{ξ}(x)| ≤ c(p, L)(1 + |ξ|)^{p−1}.

Moreover, by Theorem 2.1 (i) there exist c_{1}, c_{2} > 0 such that for a.e. x ∈ Ω and for every ξ ∈ ℝ^{N}

f(x, ξ) ≥ c_{1}|ξ|^{p} − c_{2} .  (3.3)

In view of the proof of Theorem 1.1, fixed B_{r}(x_{0}) ⊂⊂ Ω we consider the functional

G_{ϑ_{0},u_{0}}(w; B_{r}) := I(w; B_{r}) + ϑ_{0} ∫_{B_{r}} |Dw − Du_{0}|^{\frac{p+1}{2}} dx ,  (3.4)

where ϑ_{0} ≥ 0 and u_{0} ∈ W^{1,p}(B_{r}). This functional will be useful to prove the regularity of local minimizers in the general case f = f(x, u, ξ). We begin by proving an integral estimate for local minimizers of G_{ϑ_{0},u_{0}} under the supplementary assumption that f is of class C^{1} with respect to ξ and convex in this variable. This assumption will be removed in Proposition 3.2.

Proposition 3.1 Consider B_{r}(x_{0}) ⊂⊂ Ω, ϑ_{0} ≥ 0, u_{0} ∈ W^{1,p}(B_{r}(x_{0})). Let G_{ϑ_{0},u_{0}} be as in (3.4), with f of class C^{1} with respect to ξ and convex in this variable, satisfying (A1)–(A3). Let u ∈ W^{1,p}(B_{r}) be a minimizer of G_{ϑ_{0},u_{0}} in its Dirichlet class u + W_{0}^{1,p}(B_{r}). Then there exists c = c(N, p, L, ν, R) such that for every ρ < r

\begin{aligned}∫_{B_{ρ}} (1 + |Du|^{p}) dx & ≤ c [(\frac{ρ}{r})^{N} + ω(r)] ∫_{B_{r}} (1 + |Du|^{p}) dx \\ & + \frac{ϑ0^{\frac{p}{p−1}}}{[ω(r)]^{\frac{1}{p−1}}} ∫_{B_{r}} |Du − Du_{0}|^{\frac{p}{2}} dx + c r^{N} . \end{aligned}  (3.5)

Proof. Let v be the minimizer in u + W_{0}^{1,p}(B_{r}(x_{0})) of the freezed functional

w ↦ ∫_{B_{r}} f(x_{0}, Dw(x)) dx .

The function ξ ↦ f(x_{0}, ξ) satisfies the assumptions of Theorem 2.2, hence v is locally Lipschitz continuous in B_{r} and there exists a constant c depending on N, p, L, R and ν such that for all ρ < r

∫_{B_{ρ}} (1 + |Dv|^{p}) dx ≤ c (\frac{ρ}{r})^{N} ∫_{B_{r}} (1 + |Dv|^{p}) dx .

(3.3), the minimality of v and (A1) imply

∫_{B_{r}} (1 + |Dv|^{p}) dx ≤ c ∫_{B_{r}} (1 + |Du|^{p}) dx ,  (3.6)

so that for any ρ < r

∫_{B_{ρ}} (1 + |Dv|^{p}) dx ≤ c (\frac{ρ}{r})^{N} ∫_{B_{r}} (1 + |Du|^{p}) dx .  (3.7)

Define

\begin{aligned}A(Du, ρ, λ) & := \{x ∈ B_{ρ} : |Du(x)| > λ\} , \\ B(Du, ρ, λ) & := \{x ∈ B_{ρ} : |Du(x)| ≤ λ\} . \end{aligned}  (3.8)

Let R_{0} and ν_{0}, depending only on p, L, R and ν, be as in Theorem 2.1 (ii). Using Lemma 2.3 and (3.7)

\begin{matrix}∫_{B_{ρ}} |Du|^{p} dx = ∫_{B(Du,ρ,R_{0})} |Du|^{p} dx + ∫_{A(Du,ρ,R_{0})} |Du|^{p} dx \\ ≤ & c ∫_{B_{ρ}}(1 + |Dv|^{p}) dx + c ∫_{A(Du,ρ,R_{0})}(1 + |Du|^{2} + |Dv|^{2})^{\frac{p−2}{2}} |Du − Dv|^{2} dx \\ ≤ & c (\frac{ρ}{r})^{N}∫_{B_{r}}(1 + |Du|^{p}) dx + c ∫_{A(Du,ρ,R_{0})}(1 + |Du|^{2} + |Dv|^{2})^{\frac{p−2}{2}} |Du − Dv|^{2} dx. \end{matrix} (3.9)

Let us estimate the last integral using the Euler equation for G_{ϑ_{0},u_{0}}. From Theorem 2.1 (ii), the convexity of f and the minimality of u we have

∫_{A(Du,ρ,R_{0})}(1 + |Du|^{2} + |Dv|^{2})^{\frac{p−2}{2}} |Du − Dv|^{2} dx
≤\frac{1}{ν_{0}} ∫_{A(Du,ρ,R_{0})}[f(x, Dv) − f(x, Du) − ⟨D_{ξ}f(x, Du), Dv − Du⟩] dx
≤\frac{1}{ν_{0}} ∫_{B_{r}} [f(x, Dv) − f(x, Du) − ⟨D_{ξ}f(x, Du), Dv − Du⟩] dx
=\frac{1}{ν_{0}} ∫_{B_{r}}[f(x, Dv) − f(x, Du) + ϑ_{0}⟨D_{ξ}(|Du − Du_{0}|^{\frac{p+1}{2}} ), Dv − Du⟩] dx
−\frac{1}{ν_{0}} ∫_{B_{r}}⟨D_{ξ}f(x, Du) + ϑ_{0}D_{ξ}(|Du − Du_{0}|^{\frac{p+1}{2}} ), Dv − Du⟩ dx
=\frac{1}{ν_{0}} ∫_{B_{r}}[f(x, Dv) − f(x, Du) + ϑ_{0}⟨D_{ξ}(|Du − Du_{0}|^{\frac{p+1}{2}} ), Dv − Du⟩] dx .

Thus, adding and subtracting ∫_{B_{r}} f(x_{0}, Dv)dx and ∫_{B_{r}} f(x_{0}, Du)dx, and using the minimality of v it follows

∫_{A(Du,ρ,R_{0})}(1 + |Du|^{2} + |Dv|^{2})^{\frac{p−2}{2}} |Du − Dv|^{2} dx
≤\frac{1}{ν_{0}} ∫_{B_{r}}[f(x, Dv) − f(x_{0}, Dv)] dx + \frac{1}{ν_{0}} ∫_{B_{r}}[f(x_{0}, Du) − f(x, Du)] dx
+\frac{ϑ_{0}}{ν_{0}} ∫_{B_{r}}⟨D_{ξ}(|Du − Du_{0}|^{\frac{p+1}{2}}), Dv − Du⟩ dx .

From (A1) and (A3)

∫_{B_{r}}[f(x, Dv) − f(x_{0}, Dv)] dx
=∫_{A(Dv,r,R)}[f(x, Dv) − f(x_{0}, Dv)] dx +∫_{B(Dv,r,R)}[f(x, Dv) − f(x_{0}, Dv)] dx
≤ ∫_{A(Dv,r,R)}ω(|x − x_{0}|) (1 + |Dv|^{p}) dx + L∫_{B(Dv,r,R)}(1 + |Dv|^{p}) dx
≤ω(r) ∫_{B_{r}} (1 + |Dv|^{p}) dx + c(N, p, L, R) r^{N}.

Analogously,

∫_{B_{r}}[f(x_{0}, Du) − f(x, Du)] dx ≤ ω(r) ∫_{B_{r}} (1 + |Du|^{p}) dx + c(N, p, L, R) r^{N}.

Therefore from (3.6) and the minimality of v there exists c, depending only on N, p, L, R and ν, such that

\begin{matrix}∫_{A(Du,ρ,R_{0})}(1 + |Du|^{2} + |Dv|^{2})^{\frac{p−2}{2}} |Du − Dv|^{2} dx \\ ≤ & c ω(r)∫_{B_{r}}(1 + |Du|^{p})dx + c ϑ_{0}∫_{B_{r}}|Du − Du_{0}|^{\frac{p−1}{2}} |Du − Dv| dx + c r^{N} . \end{matrix}

This inequality, together with (3.9) and Young inequality, implies that for every ρ < r there exists c, depending on N, p, L, R and ν, such that

\begin{matrix}∫_{B_{ρ}} |Du|^{p} dx ≤ c [(\frac{ρ}{r})^{N} + ω(r)] ∫_{B_{r}} (1 + |Du|^{p}) dx \\ + & c ω(r) ∫_{B_{r}} |Du − Dv|^{p} dx + c\frac{ϑ0^{\frac{p}{p−1}}}{[ω(r)]^{\frac{1}{p−1}}} ∫_{B_{r}} |Du − Du_{0}|^{\frac{p}{2}} dx + c r^{N} \end{matrix}

and from (3.6) the thesis follows.

An approximation argument allows us to remove the differentiability assumption on f.

Proposition 3.2 Consider B_{r}(x_{0}) ⊂⊂ Ω, ϑ_{0} ≥ 0, u_{0} ∈ W^{1,p}(B_{r}(x_{0})). Let G_{ϑ_{0},u_{0}} be as in (3.4), with f convex with respect to ξ, satisfying (A1)–(A3). Then for every ρ < r

∫_{B_{ρ}} |Du|^{p}dx ≤ c [(\frac{ρ}{r})^{N} + ω(r)] ∫_{B_{r}}(1 + |Du|^{p} + |Du_{0}|^{p}) dx

+ c \frac{ϑ0^{\frac{2p}{p−1}}}{[ω(r)]^{\frac{p+1}{p−1}}} r^{N} + c r^{N},

with c depending on N, p, L, R and ν.

Proof. Let σ ∈ C_{c}^{∞}(B_{1}(0), [0, +∞)) be a radially symmetric mollifier such that ∫_{B_{1}(0)} σ(z) dz = 1, and define

f_{h}(x, ξ) := ∫_{B_{1}(0)} σ(z)f(x, ξ + \frac{1}{h}z) dz .

(f_{h}) satisfies the following assumptions:

(H1) there exists \overline{L}, depending on L and p, but not on h, such that for every h, for a.e. x ∈ Ω and for every ξ ∈ ℝ^{N}

0 ≤ f_{h}(x, ξ) ≤ \overline{L}(1 + |ξ|^{p}) ;

(H2) there exists ν_{1} > 0, depending on p and ν, but not on h, such that if [ξ, η] ⊂ ℝ^{N} \ B_{R+1}(0), then for a.e. x ∈ Ω

f_{h}(x, \frac{ξ + η}{2} ) ≤ \frac{1}{2}f_{h}(x, ξ) + \frac{1}{2}f_{h}(x, η) − ν_{1}(1 + |ξ|^{2} + |η|^{2})^{\frac{p−2}{2}} |ξ − η|^{2} ;

(H3) there exists c = c(p) such that for every x, y ∈ Ω and for every ξ with |ξ| ≥ R + 1

|f_{h}(x, ξ) − f_{h}(y, ξ)| ≤ c ω(|x − y|)(1 + |ξ|^{p}) .

Moreover, from (3.3) there exist \overline{c}_{1}, \overline{c}_{2} > 0, independent of h, such that

f_{h}(x, ξ) ≥ \overline{c}_{1}|ξ|^{p} − \overline{c}_{2}, ∀(x, ξ) ∈ Ω × ℝ^{N} .  (3.10)

Let u_{h} be a minimizer in u + W_{0}^{1,p}(B_{r}) of

G_{h}(w; B_{r}) := ∫_{B_{r}} f_{h}(x, Dw) dx + ϑ_{0} ∫_{B_{r}} |Dw − Du_{0}|^{\frac{p+1}{2}} dx .

Using (3.10), the minimality of u_{h}, (H1) and Young inequality, it is not difficult to prove that there exists c = c(N, p, L, R, ν, ϑ_{0}), such that

∫_{B_{r}} |Du_{h}|^{p} dx ≤ c ∫_{B_{r}}(1 + |Dw|^{p} + |Du_{0}|^{\frac{p+1}{2}} ) dx ,

for every w ∈ u + W_{0}^{1,p}(B_{r}). In particular

∫_{B_{r}} |Du_{h}|^{p} dx ≤ c ∫_{B_{r}}(1 + |Du|^{p} + |Du_{0}|^{p}) dx  (3.11)

and u_{h} is a Q-minimizer of

w ↦ ∫_{B_{r}} (1 + |Dw|^{p} + |Du_{0}|^{\frac{p+1}{2}} ) dx .

Therefore, from Lemma 2.4 there exist τ > 1 and c > 0 such that u_{h} ∈ W_{loc}^{1,pτ} (B_{r}) for any h and for every B_{ρ}(x_{1}) ⊂ B_{r}

(− ∫_{B_{ρ/2}(x_{1})} |Du_{h}|^{pτ} dx)^{\frac{1}{τ}} ≤ c − ∫_{B_{ρ}(x_{1})}(1 + |Du_{h}|^{p} + |Du_{0}|^{p}) dx .

This inequality, together with (3.11), implies that for every ρ < r there exists c = c(ρ, r) such that

(∫_{B_{ρ}} |Du_{h}|^{pτ} dx)^{\frac{1}{τ}} ≤ c ∫_{B_{r}}(1 + |Du|^{p} + |Du_{0}|^{p}) dx .  (3.12)

Moreover, (3.11) implies that the sequence (u_{h}) is bounded in W^{1,p}(B_{r}). Up to a subsequence, we may assume that there exists u_{∞} ∈ u + W_{0}^{1,p}(B_{r}) such that u_{h} ⇀ u_{∞} in the weak topology of W^{1,p}(B_{r}). Let us prove that u_{∞} is a minimizer of G_{ϑ_{0},u_{0}} defined in (3.4). We shall use the notations in (3.8). Since G_{ϑ_{0},u_{0}} is lower semicontinuous with respect to the weak topology of W^{1,p}, then for every ρ < r and k ∈ ℕ

G_{ϑ_{0},u_{0}}(u_{∞}; B_{ρ}) ≤ lim inf_{h→∞} ∫_{B_{ρ}} [f(x, Du_{h}) + ϑ_{0}|Du_{h} − Du_{0}|^{\frac{p+1}{2}} ] dx

≤ lim sup_{h→∞} ∫_{A(Du_{h},ρ,k)} f(x, Du_{h}) dx

+ lim sup_{h→∞} \{∫_{B(Du_{h},ρ,k)}f(x, Du_{h})dx + ϑ_{0} ∫_{B_{ρ}}|Du_{h} − Du_{0}|^{\frac{p+1}{2}} dx\} .

(3.13)

Since f is convex with respect to ξ and (A1) holds, then for a.e. x ∈ Ω and for any ξ and η in ℝ^{N},

|f(x, ξ) − f(x, η)| ≤ c(p, L)(1 + |ξ| + |η|)^{p−1}|ξ − η|;

therefore

lim_{h→∞} ∫_{B(Du_{h},ρ,k)} |f(x, Du_{h}) − f_{h}(x, Du_{h})| dx

≤c lim_{h→∞} \frac{1}{h} ∫_{B(Du_{h},ρ,k)}(1 + |Du_{h}|)^{p−1} dx
≤c lim_{h→∞} \frac{1}{h}(1 + k)^{p−1}|B(Du_{h}, ρ, k)| = 0 .

(3.14)

Thus, from (3.13) and (3.14)

G(u_{∞}; B_{ρ}) ≤ lim sup_{h→∞} ∫_{A(Du_{h},ρ,k)} f(x, Du_{h}) dx

+ lim sup_{h→∞} ∫_{B_{r}} [f_{h}(x, Du_{h}) + ϑ_{0}|Du_{h} − Du_{0}|^{\frac{p+1}{2}} ] dx .

From (A1), Hölder inequality, (3.12) and the minimality of u_{h} we get that

G_{ϑ_{0},u_{0}}(u_{∞}; B_{ρ}) ≤ L lim sup_{h→∞} \{(∫_{B_{ρ}}(1 + |Du_{h}|^{\frac{p}{τ}})dx)^{\frac{1}{τ}}|A(Du_{h}, ρ, k)|^{\frac{τ−1}{τ}} \}

+ lim sup_{h→∞} ∫_{B_{r}}[f_{h}(x, Du) + ϑ_{0}|Du − Du_{0}|^{\frac{p+1}{2}} ] dx

≤c k^{p(1−τ)} lim sup_{h→∞} ∫_{B_{ρ}}(1 + |Du_{h}|^{pτ}) dx +∫_{B_{r}}[f(x, Du) + ϑ_{0}|Du − Du_{0}|^{\frac{p+1}{2}} ] dx .

The last inequality and (3.12) imply

G_{ϑ_{0},u_{0}}(u_{∞}; B_{ρ}) ≤ c k^{p(1−τ)} + G_{ϑ_{0},u_{0}}(u; B_{r}) ,

where c = c(N, p, L, u, u_{0}, ϑ_{0}, ρ, r). Letting k go to infinity and then ρ go to r we infer G_{ϑ_{0},u_{0}}(u_{∞}; B_{r}) = G_{ϑ_{0},u_{0}}(u; B_{r}), so that u_{∞} is a minimizer of G_{ϑ_{0},u_{0}}(w; B_{r}).

Now we compare u_{∞} with u. For every x ∈ B_{r} we apply Theorem 2.1 (ii) with ξ = \frac{1}{2}(Du(x) + Du_{∞}(x)) and η = Du(x). Thus, from (3.2) we have

\begin{matrix}∫_{A(Du+Du_{∞},r,2R_{0})} [f(x, Du) − f (x, \frac{Du + Du_{∞}}{2} )] dx \\ ≥ & ∫_{A(Du+Du_{∞},r,2R_{0})} ⟨q_{ξ}(x), \frac{Du − Du_{∞}}{2} ⟩ dx \\ + & ν ∫_{A(Du+Du_{∞},r,2R_{0})} (1 + |\frac{Du + Du_{∞}}{2} |^{2} + |Du|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx. \end{matrix} (3.15)

Analogously, Theorem 2.1 (ii) with η = Du_{∞}(x) yields

\begin{matrix}∫_{A(Du+Du_{∞},r,2R_{0})} [f(x, Du_{∞}) − f (x, \frac{Du + Du_{∞}}{2} )] dx \\ ≥ & ∫_{A(Du+Du_{∞},r,2R_{0})} ⟨q_{ξ}(x), \frac{Du_{∞} − Du}{2}⟩ dx \\ + & ν ∫_{A(Du+Du_{∞},r,2R_{0})}(1 +|\frac{Du + Du_{∞}}{2} |^{2}+ |Du_{∞}|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx; \end{matrix} (3.16)

adding (3.15) and (3.16)

\begin{matrix}∫_{A(Du+Du_{∞},r,2R_{0})} [f(x, Du) + f(x, Du_{∞})] dx \\ ≥ & 2∫_{A(Du+Du_{∞},r,2R_{0})} f (x, \frac{Du + Du_{∞}}{2} ) dx \\ + & ν ∫_{A(Du+Du_{∞},r,2R_{0})} (1 + |\frac{Du + Du_{∞}}{2} |^{2} + |Du|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx \\ + & ν ∫_{A(Du+Du_{∞},r,2R_{0})}(1 + |\frac{Du + Du_{∞}}{2} |^{2}+ |Du_{∞}|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx. \end{matrix} (3.17)

By convexity

∫_{B(Du+Du_{∞},r,2R_{0})} f(x, \frac{Du + Du_{∞}}{2}) dx +ϑ_{0}∫_{B_{r}} |\frac{Du + Du_{∞}}{2} − Du_{0}|^{\frac{p+1}{2}} dx

≤\frac{1}{2} ∫_{B(Du+Du_{∞},r,2R_{0})} f(x, Du) dx + \frac{ϑ_{0}}{2} ∫_{B_{r}} |Du − Du_{0}|^{\frac{p+1}{2}} dx
+\frac{1}{2} ∫_{B(Du+Du_{∞},r,2R_{0})} f(x, Du_{∞}) dx + \frac{ϑ_{0}}{2} ∫_{B_{r}} |Du_{∞} − Du_{0}|^{\frac{p+1}{2}} dx ,

therefore, from (3.17) we get

G_{ϑ_{0},u_{0}} (\frac{u + u_{∞}}{2} ; B_{r}) ≤ \frac{1}{2}G_{ϑ_{0},u_{0}}(u; B_{r}) + \frac{1}{2}G_{ϑ_{0},u_{0}}(u_{∞}; B_{r})

−\frac{ν}{2} ∫_{A(Du+Du_{∞},r,2R_{0})}(1 + |\frac{Du + Du_{∞}}{2} |^{2}+ |Du|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx
−\frac{ν}{2} ∫_{A(Du+Du_{∞},r,2R_{0})}(1 + |\frac{Du + Du_{∞}}{2} |^{2}+ |Du_{∞}|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx.

The minimality of u_{∞}, together with the previous inequality, yields

G_{ϑ_{0},u_{0}} (\frac{u + u_{∞}}{2} ; B_{r}) ≤ G_{ϑ_{0},u_{0}}(u; B_{r})

−\frac{ν}{2} ∫_{A(Du+Du_{∞},r,2R_{0})} (1 + |\frac{Du + Du_{∞}}{2} |^{2}+ |Du|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx
−\frac{ν}{2} ∫_{A(Du+Du_{∞},r,2R_{0})}(1 + |\frac{Du + Du_{∞}}{2} |^{2}+ |Du_{∞}|^{2})^{\frac{p−2}{2}} |\frac{Du − Du_{∞}}{2} |^{2} dx,

which implies that the Lebesgue measure of the set

{x ∈ B_{r}:|Du(x) + Du_{∞}(x)| > 2R_{0}} ∩ {x ∈ B_{r}:|Du(x) − Du_{∞}(x)| > 0} is zero. Therefore, for every ρ < r

∫_{B_{ρ}} |Du|^{p} dx ≤ c ∫_{B_{ρ}} |Du_{∞}|^{p} dx + c ∫_{B_{ρ}} |Du + Du_{∞}|^{p} dx

≤c ∫_{B_{ρ}} |Du_{∞}|^{p} dx + c 2^{p}R_{0}^{p} ρ^{N} + c ∫_{A(Du+Du_{∞},ρ,2R_{0})} |Du + Du_{∞}|^{p} dx
=c ∫_{B_{ρ}}|Du_{∞}|^{p} dx + c ρ^{N} + c ∫_{A(Du+Du_{∞},ρ,2R_{0})∩{|Du−Du_{∞}|=0}}|Du_{∞}|^{p} dx ,

thus there exists c = c(N, p, L, R, ν) such that

∫_{B_{ρ}} |Du|^{p} dx ≤ c ∫_{B_{ρ}}(1 + |Du_{∞}|^{p}) dx .  (3.18)

Let us estimate the right-hand side. Since f_{h} is of class C^{1} with respect to ξ and satisfies (H1)–(H3), from Proposition 3.1 estimate (3.5) holds with u replaced by u_{h}. Hence, for all ρ < r,

∫_{B_{ρ}} |Du_{∞}|^{p} dx ≤ lim inf_{h→∞} ∫_{B_{ρ}}(1 + |Du_{h}|^{p}) dx

≤ lim sup_{h→∞} \{c [(\frac{ρ}{r})^{N} + ω(r)]∫_{B_{r}}(1 + |Du_{h}|^{p}) dx + c r^{N}\}
+ lim sup_{h→∞} \frac{c ϑ0^{\frac{p}{p−1}}}{[ω(r)]^{\frac{1}{p−1}}} ∫_{B_{r}} |Du_{h} − Du_{0}|^{\frac{p}{2}} dx,

where c is independent of h, ρ and r. Young inequality implies

\frac{cϑ0^{\frac{p}{p−1}}}{[ω(r)]^{\frac{1}{p−1}}} ∫_{B_{r}} |Du_{h} − Du_{0}|^{\frac{p}{2}} dx

=\frac{cϑ0^{\frac{p}{p−1}}}{[ω(r)]^{\frac{1}{p−1}}} ∫_{B_{r}} [\frac{ϑ_{0}}{ω(r)}]^{\frac{p}{2(p−1)}} [\frac{ω(r)}{ϑ_{0}}]^{\frac{p}{2(p−1)}} |Du_{h} − Du_{0}|^{\frac{p}{2}} dx

≤c ω(r) ∫_{B_{r}} |Du_{h} − Du_{0}|^{p} dx + c\frac{ϑ0^{\frac{2p}{p−1}}}{[ω(r)]^{\frac{p+1}{p−1}}} r^{N}

≤c ω(r) ∫_{B_{r}}(|Du_{h}|^{p} + |Du_{0}|^{p}) dx + c\frac{ϑ0^{\frac{2p}{p−1}}}{[ω(r)]^{\frac{p+1}{p−1}}} r^{N} .

Therefore, from (3.11), there exists c, depending only on N, p, L, R and ν, such that for every ρ < r

\begin{aligned}∫_{B_{ρ}} |Du_{∞}|^{p}dx & ≤ c [(\frac{ρ}{r})^{N} + ω(r)] ∫_{B_{r}}(1 + |Du|^{p} + |Du_{0}|^{p}) dx \\ & + c\frac{ϑ0^{\frac{2p}{p−1}}}{[ω(r)]^{\frac{p+1}{p−1}}} r^{N} + c r^{N}. \end{aligned}

Finally, by (3.18) the thesis follows.

If ϑ_{0} = 0 we get a regularity result for local minimizers of the functional I in (3.1) when f is convex in ξ.

Theorem 3.3 Let u be a local minimizer of the functional I, whose integrand f(x, ξ) is a Carathéodory function, convex with respect to the last variable and satisfies (A1)–(A3). Then u is locally in C^{0,α}(Ω) for all α < 1. Moreover, for all α < 1 there exists c > 0, depending on N, p, L, R, ν and α, such that for every B_{r} ⊂⊂ Ω and ρ < r

∫_{B_{ρ}} |Du|^{p} dx ≤ c (\frac{ρ}{r})^{N−p+pα} ∫_{B_{r}} (1 + |Du|^{p}) dx .  (3.19)

Proof. When ϑ_{0} = 0, Proposition 3.2 implies that for every ρ < r

∫_{B_{ρ}} |Du|^{p} dx ≤ c [(\frac{ρ}{r})^{N} + ω(r)] ∫_{B_{r}}(1 + |Du|^{p}) dx + c r^{N} .

A standard iteration argument (see [11], p.170) leads to estimate (3.19). The Hölder continuity of u follows from a characterization of Campanato spaces (see [16], p.57).

4 Proof of Theorem 1.1

In this section we prove Theorem 1.1. First we shall suppose that f is convex with respect to ξ. This supplementary assumption will be removed observing that a local minimizer of F is a local minimizer for the relaxed functional, too.

The Ekeland variational principle enables us to compare the local minimizer of F with a minimizer of a functional whose integrand does not explicitly depend on u.

Theorem 4.1 (Ekeland variational principle [4]) Let (V, d) be a complete metric space and let H : V → (−∞, +∞] be a lower semicontinuous functional with finite infimum. If u ∈ V is such that H(u) ≤ inf_{V} H + ϵ for some ϵ > 0, then there exists v_{0} ∈ V such that

(i) d(u, v_{0}) ≤ 1 ,

(ii) H(v_{0}) ≤ H(u) ,

(iii) v_{0} minimizes in V the functional E(w) := H(w) + ϵ d(v_{0}, w).

Proof of Theorem 1.1. Without loss of generality we assume ω in (A3) to be a concave function. From Theorem 2.1 (i) a local minimizer of F is a Q-minimizer of (2.2) with ϕ = 0, therefore from Lemma 2.4 there exists q > p such that |Du| ∈ L_{loc}^{q}(Ω). Since we aim to prove a local result, we can assume that u ∈ W^{1,q}(Ω) and that for any ball B_{r}(x_{1}) ⊂ Ω

[− ∫_{B_{r/2}(x_{1})} |Du|^{q} dx]^{\frac{1}{q}} ≤ c [− ∫_{B_{r}(x_{1})}(1 + |Du|^{p}) dx]^{\frac{1}{p}} .  (4.1)

Moreover (see [14]) we can assume u ∈ C^{0,γ}(Ω) for some 0 < γ < 1. We denote by [u]_{γ} the Hölder constant of u in Ω. Let us fix B_{r}(x_{0}) such that B_{4r}(x_{0}) ⊂⊂ Ω.

Step 1. Assume that f is convex in ξ. Consider the space V = u + W_{0}^{1, \frac{p+1}{2}} (B_{r}) and let v be the minimizer in V of the functional

H(w; B_{r}) := ∫_{B_{r}} h(x, Dw(x)) dx ,  (4.2)

where

h(x, ξ) := f(x, u(x), ξ).  (4.3)

By Theorem 2.1 (i), the minimality of v and (A1)

∫_{B_{r}} |Dv|^{p} dx ≤ c ∫_{B_{r}}(1 + |Du|^{p}) dx .  (4.4)

Using notations (3.8), the minimality of u and (A3) we have

\begin{matrix}H(u; B_{r}) = F(u; B_{r}) ≤ F(v; B_{r}) \\ = & H(v; B_{r}) + ∫_{B(Dv,r,R)} [f(x, v, Dv) − f(x, u, Dv)] dx \\ + & ∫_{A(Dv,r,R)} [f(x, v, Dv) − f(x, u, Dv)] dx \\ ≤ & H(v; B_{r}) + L ∫_{B(Dv,r,R)}(1 + |Dv|^{p}) dx + ∫_{A(Dv,r,R)} ω(|v − u|)(1 + |Dv|^{p})dx \\ ≤ & inf_{V} H(w; B_{r}) + ∫_{B_{r}} ω(|v − u|)(1 + |Dv|^{p}) dx + c r^{N}. \end{matrix} (4.5)

Let s ∈ (p, q) be the exponent of the higher integrability of Dv which follows from Lemma 2.5. Using the Hölder inequality, (4.1), the boundedness and the concavity of ω, we estimate the last integral in (4.5) as follows:

\begin{matrix}∫_{B_{r}} ω(|v − u|)(1 + |Dv|^{p}) dx \\ ≤ & c |B_{r}| (− ∫_{B_{r}}(1 + |Dv|^{s}) dx)^{\frac{p}{s}} (− ∫_{B_{r}} ω^{\frac{s}{s−p}} (|v − u|) dx)^{1− \frac{p}{s}} \\ ≤ & c |B_{r}| (− ∫_{B_{2r}}(1 + |Du|^{q}) dx)^{\frac{p}{q}} (− ∫_{B_{r}} ω(|v − u|) dx)^{1− \frac{p}{s}} \\ ≤ & c |B_{r}| − ∫_{B_{4r}}(1 + |Du|^{p}) dx · ω^{1− \frac{p}{s}} (− ∫_{B_{r}} |v − u| dx) \\ ≤ & c ∫_{B_{4r}}(1 + |Du|^{p}) dx · ω^{σ}(− ∫_{B_{r}} |v − u| dx) , \end{matrix}  (4.6)

with σ = 1 − \frac{p}{s}. Caccioppoli inequality for the minimizer u (see e.g. [14]) gives

∫_{B_{ρ}(x_{1})} |Du|^{p} dx ≤ c ∫_{B_{2ρ}(x_{1})} (1 + \frac{|u − u_{x_{1},2ρ}|^{p}}{ρ^{p}} ) dx ,

which holds for every B_{2ρ}(x_{1}) ⊆ B_{2r}(x_{0}) (here u_{x_{1},ρ} stands for − ∫_{B_{ρ}(x_{1})} u(x)dx). Thus, using the Poincaré inequality and (4.4), we get that

\begin{aligned}& − ∫_{B_{r}} |v − u| dx ≤ (c r^{p} − ∫_{B_{r}} |Dv − Du|^{p} dx)^{\frac{1}{p}} ≤ (c r^{p} − ∫_{B_{r}}(1 + |Du|^{p}) dx)^{\frac{1}{p}} \\ ≤ & [c r^{p} − ∫_{B_{2r}} (1 + \frac{|u − u_{x_{0},2r}|^{p}}{r^{p}} ) dx]^{\frac{1}{p}} ≤ (c r^{p} + c [u]_{γ}^{p} r^{pγ})^{\frac{1}{p}} ≤ \overline{c} r^{γ}, \end{aligned}

where \overline{c} depends also on [u]_{γ}; then

ω^{σ}(− ∫_{B_{r}} |v − u|dx) ≤ ω^{σ}(\overline{c}r^{γ}) .

This estimate, together with (4.5), (4.6) and (4.4), leads to

H(u; B_{r}) ≤ inf_{V} H(w; B_{r}) + c ω^{σ} (\overline{c}r^{γ}) ∫_{B_{4r}}(1 + |Du|^{p})dx + c r^{N} .  (4.7)

Step 2. Let us define

H(r) := c ω^{σ}(\overline{c}r^{γ}) ∫_{B_{4r}}(1 + |Du|^{p}) dx + c r^{N}  (4.8)

and apply Theorem 4.1 with V endowed with the distance

d(w_{1}, w_{2}) := [\frac{H(r)}{r^{N}}]^{− \frac{p+1}{2p}} r^{−N} ∫_{B_{r}} |Dw_{1} − Dw_{2}|^{\frac{p+1}{2}} dx ,  (4.9)

and H defined in (4.2). Thus, by (4.7), there exists v_{0} ∈ V such that

∫_{Br} |Du − Dv_{0}|^{\frac{p+1}{2}} dx ≤ [\frac{H(r)}{r^{N}}]^{\frac{p+1}{2p}} r^{N},  (4.10)
H(v_{0}; B_{r}) ≤ H(u; B_{r}) ,  (4.11)
v_{0} is a minimizer of E(w; B_{r}) in V ,

where

E(w; B_{r}) := H(w; B_{r}) + [\frac{H(r)}{r^{N}}]^{\frac{p−1}{2p}} ∫_{B_{r}} |Dw − Dv_{0}|^{\frac{p+1}{2}} dx .  (4.12)

The minimality of v_{0} implies that for every φ ∈ W_{0}^{1,p}(B_{r})

H(v_{0}; spt φ) ≤ H(v_{0} + φ; spt φ) + [\frac{H(r)}{r^{N}}]^{\frac{p−1}{2p}}∫_{spt φ}|D(v_{0} + φ) − Dv_{0}|^{\frac{p+1}{2}} dx.

Using Young inequality and noting that (\frac{2p}{p+1})′ = \frac{2p}{p−1}, we obtain

H(v_{0}; spt φ) ≤ H(v_{0} + φ; spt φ) + \frac{c_{1}}{2} ∫_{spt φ} |Dv_{0}|^{p} dx

+ c ∫_{spt φ} |Dv_{0} + Dφ|^{p} dx + c \frac{H(r)}{r^{N}}|spt φ| .

From this inequality, Theorem 2.1 (i) and (A1), it follows that

∫_{spt φ} |Dv_{0}|^{p} dx ≤ c ∫_{spt φ} (1 + |Dv_{0} + Dφ|^{p} + \frac{H(r)}{r^{N}}) dx ,

therefore v_{0} is a Q-minimizer (with Q depending on N, p, L, R and ν) of the functional

w ↦ ∫_{B_{r}} (1 + |Dw|^{p} + \frac{H(r)}{r^{N}}) dx .

Thus, there exist λ ∈ (p, q) and c > 0, independent of v_{0}, such that

\begin{aligned}(− ∫_{B_{r/2}} |Dv_{0}|^{λ} dx)^{\frac{p}{λ}} & ≤ c − ∫_{B_{r}}(1 + |Dv_{0}|^{p}) dx + c [1 + \frac{H(r)}{r^{N}}] \\ & ≤ c − ∫_{B_{4r}}(1 + |Du|^{p}) dx , \end{aligned}  (4.13)

where the last inequality follows from (4.8), Theorem 2.1 (i), (4.11) and (A1). Notice that h defined in (4.3) satisfies assumptions (A1)–(A3), with a slightly different modulus of continuity. Precisely, for any 0 < α < 1, h satisfies (A3) with ω replaced by

ω̃_{α}(t) := max \{[ω^{σ} (\overline{c}t^{γ})]^{\frac{p−1}{2p}} , ω(t + [u]_{γ}t^{γ}) , t^{p(1−α) \frac{p−1}{p+1}} \} .  (4.14)

Apply Proposition 3.2 to E(w; B_{r}) defined in (4.12), with ϑ_{0} and u_{0} replaced by

[\frac{H(r)}{r^{N}}]^{\frac{p−1}{2p}} and v_{0}, respectively. Then for every ρ ≤ r/2

\begin{matrix}∫_{B_{ρ}} |Du|^{p}dx ≤ 2^{p−1} ∫_{B_{ρ}} |Dv_{0}|^{p} dx + 2^{p−1} ∫_{B_{r/2}} |Du − Dv_{0}|^{p} dx \\ ≤ & c [(\frac{ρ}{r})^{N} + ω̃_{α}(r)] ∫_{B_{r}}(1 + |Dv_{0}|^{p}) dx \\ + & c\frac{H(r)}{[ω̃_{α}(r)]^{\frac{p+1}{p−1}}} + c r^{N} + 2^{p−1} ∫_{B_{r/2}} |Du − Dv_{0}|^{p} dx \\ ≤ & c [(\frac{ρ}{r})^{N} + ω̃_{α}(r)] ∫_{B_{r}}(1 + |Du|^{p}) dx + c\frac{ω^{σ} (\overline{c}r^{γ})}{[ω̃_{α}(r)]^{\frac{p+1}{p−1}}}∫_{B_{4r}}(1 + |Du|^{p}) dx \\ + & c r^{N−p+pα} + c r^{N} + 2^{p−1}∫_{B_{r/2}} |Du − Dv_{0}|^{p} dx \\ ≤ & c [(\frac{ρ}{r})^{N}+ ω̃_{α}(r)] ∫_{B_{4r}}(1 + |Du|^{p})dx+c r^{N−p+pα} + 2^{p−1} ∫_{B_{r/2}}|Du − Dv_{0}|^{p}dx. \end{matrix} (4.15)

In order to estimate the last integral, let ϑ ∈ (0, 1) be such that \frac{ϑ}{λ} + \frac{2(1−ϑ)}{p+1} = \frac{1}{p}, where λ is the exponent in (4.13). Using (4.1), (4.13), (4.10) and (4.11), we get

\begin{matrix}∫_{B_{r/2}} |Du − Dv_{0}|^{p}dx \\ ≤ & c|B_{r}| (− ∫_{B_{r/2}} |Du − Dv_{0}|^{λ} dx)^{ϑ \frac{p}{λ}} (− ∫_{B_{r}} |Du − Dv_{0}|^{\frac{p+1}{2}} dx)^{(1−ϑ) \frac{2p}{p+1}} \\ ≤ & c r^{N} (− ∫_{B_{4r}}(1 + |Du|^{p}) dx)^{ϑ} [\frac{H(r)}{r^{N}}]^{1−ϑ} \\ ≤ & c [ω^{σ} (\overline{c}r^{γ})]^{1−ϑ} ∫_{B_{4r}}(1 + |Du|^{p}) dx + c r^{N(1−ϑ)} (∫_{B_{4r}}(1 + |Du|^{p}) dx)^{ϑ} . \end{matrix} (4.16)

Finally, collecting (4.15), (4.16) and using Young inequality, we deduce that for every ϵ > 0 there exists c(ϵ) such that

\begin{matrix}∫_{B_{ρ}} |Du|^{p}dx ≤ c [(\frac{ρ}{r})^{N} + ω̃_{α}(r) + ϵ] ∫_{B_{4r}}(1 + |Du|^{p}) dx \\ + & c[ω̃_{α}(r)]^{(1−ϑ) \frac{2p}{p−1}} ∫_{B_{4r}}(1 + |Du|^{p}) dx + c r^{N−p+pα} + c(ϵ) r^{N}, \end{matrix}

which implies

∫_{B_{ρ}} |Du|^{p} dx ≤ c [(\frac{ρ}{r})^{N} + [ω̃_{α}(r)]^{δ} + ϵ] ∫_{B_{4r}}(1 + |Du|^{p}) dx + c_{ϵ}r^{N−p+pα},

for a certain δ > 0 independent of ρ and r. From this inequality the thesis follows by Lemma 2.6 and by a standard iteration argument (see [11], p.170).

Step 3. Now we consider a general f non convex with respect to the last variable. We observe that local minimizers of F are also local minimizers of the relaxed functional

F^{∗∗}(u; B_{r}(x_{0}))= inf{lim inf_{h→+∞}∫_{B_{r}(x_{0})}f(x, u_{h}, Du_{h}) dx : u_{h} ⇀ u in W_{0}^{1,p}(B_{r}(x_{0}))}

which is equal to

∫_{B_{r}(x_{0})} f^{∗∗}(x, u(x), Du(x)) dx ,

where ξ ↦ f^{∗∗}(x, u, ξ) is the bipolar of ξ ↦ f(x, u, ξ). The result follows immediately from the fact that by Theorem 2.1 (ii) there exists R_{0} such that

f(x, u, ξ) = f^{∗∗}(x, u, ξ)

in Ω × ℝ × (ℝ^{N} \ B_{R_{0}}(0)).

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