Hölder Continuity up to the Boundary of Minimizers for Some Integral Functionals with Degenerate

defined in a suitable weighted Banach space; Ω is an open and bounded set of Rn and ∇2u= {Dαu : |α| = 1,2}. We note that in this paper we obtain our regularity result directly working with the functional I(u) instead of working with its Euler equation. In fact, we will not suppose any differentiability of A(x,ξ), principal part of integrand of the functional I(u), but only that it is a Carathèodory function, convex with respect to ξ, satisfying the following growth condition: for almost every x ∈Ω and for every ξ = {ξα : |α| = 1,2},


Introduction
In this paper, we will study regularity properties of minimizers for integral functionals of the form (1.1) defined in a suitable weighted Banach space; Ω is an open and bounded set of R n and ∇ 2 u = {D α u : |α| = 1,2}. We note that in this paper we obtain our regularity result directly working with the functional I(u) instead of working with its Euler equation. In fact, we will not suppose any differentiability of A(x,ξ), principal part of integrand of the functional I(u), but only that it is a Carathèodory function, convex with respect to ξ, satisfying the following growth condition: for almost every x ∈ Ω and for every ξ = {ξ α : |α| = 1,2}, 2 Journal of Applied Mathematics and Stochastic Analysis where c 1 ,c 2 are positive constants, f (x) is a nonnegative function, belonging to a suitable Lebesgue space, and ν(x), μ(x) are positive measurable functions that we will specify later. This kind of condition, introduced by Skrypnik in [1], is stronger than the one that is usually considered (see, e.g., [2,3]), but the usual growth condition in general cannot give even the boundedness of the minima of I(u) (see [4]).
In [5], boundedness and Hölder continuity for minimizers of the same functional I(u) in the interior of Ω were already established. Now the aim of this paper is to establish Hölder continuity up to the boundary of any minimizer u(x).
Under some hypotheses on weighted functions in order to guarantee embedding between Banach spaces and under some hypotheses of regularity of the boundary ∂Ω, using the convexity properties of the functions A(x,ξ) and A 0 (x,η) and the above growth conditions, we obtain an integral estimate of the gradient of the minimizers. Then the iterative Moser method (see [6]) opportunely modified permits us to estimate the oscillation of u(x) near the boundary of Ω. So with the interior regularity result of [5], we obtain our goal.
Note that in the case of 2p < q < n, some results on Hölder continuity of solutions of equations and variational inequalities with degenerate nonlinear high-order operators have been obtained in [1,[13][14][15][16].

Hypotheses and statement of main results
In this section, we give hypotheses concerning weighted functions in order to define our weighted Banach spaces, and to guarantee some embedding results, we give hypotheses on the integrand functions and state the main result.
Let Ω be a bounded open set of R n . Let p ≥ 2, q be two real numbers such that 2p < q < n. Hypothesis 2.1. Let ν(x) : Ω→R + be a measurable function such that is the space of all functions u ∈ L q (Ω) such that their derivatives, in the sense of distribution, D α u, |α| = 1, are functions for which the following properties hold: 2,p (Ω,ν,μ) is the space of all functions u ∈ W 1,q (Ω,ν) such that their derivatives, in the sense of distribution, D α u,|α| = 2, are functions for which the following properties hold: μ 1/p D α u ∈ L p (Ω) if |α| = 2; W 1,q 2,p (ν,μ,Ω) is a Banach space with respect to the norm Hypothesis 2.3. We assume the function 1/ν ∈ L t (Ω) with t > n/q.
(2.6) Hypothesis 2.5. There exists a constant c > 0 such that for every y ∈ Ω and ρ > 0, with B(y, ρ) ⊂ Ω, we have We need these previous hypotheses in order to ensure the regularity of minimizers of our functional in the interior of Ω. To have the regularity to the boundary, we need the following further hypotheses concerning the boundary of Ω and the extension of weights on the boundary. Hypothesis 2.6. There exist c * , ρ * such that for every y ∈ ∂Ω and ρ ∈ ]0,ρ * [, we have meas B(y,ρ) \ Ω ≥ c * meas B(y,ρ) . (2.8) Consequently, Ω belongs to the class S (see, e.g., [17]). Let us put Hypothesis 2.7. There exist a positive measurable function ν(x) : Ω→R and a real positive (2.10) We denote by R n,2 the space of all sets ξ = {ξ α ∈ R : |α| = 1,2} of real numbers.
Hypothesis 2.9. There exist c 1 ,c 2 > 0 and a nonnegative function f ∈ L t * (Ω) such that for almost x ∈ Ω and for every ξ ∈ R n,2 , the inequality (1.2) holds.
such that, almost everywhere in Ω and for all η ∈ R, the following inequality holds: Let I : From the theory of monotone and coercive operators, it is well known that under the previous hypotheses there exists u(x) minimizer of I in • W 1,q 2,p (Ω,ν,μ). Moreover, u(x) is essentially bounded in Ω and Hölder continuous in every compact subset of Ω (see [5]). Now, we can formulate our regularity result more precisely.
in Ω, and for every x, y ∈ Ω, we have where positive constants C and γ depend only on known values and on u L q (Ω) .
We will assume that It is known that there exists a set E ⊂ Ω ∩ B(y,2ρ) such that measE = 0, and for all (3.7) We introduce now the following function: 6 Journal of Applied Mathematics and Stochastic Analysis and the cutoff function ϕ ∈ C ∞ (Ω), 0 ≤ ϕ ≤ 1 in Ω, defined by 2ρ).
We observe that if (3.6) does not hold, it is possible to repeat all considerations substi- Let us fix s > m 1 and define (3.10) It is useful to note that due to (3.6) and (3.7) F(x) (3.13) Next, if we put it follows that 0 ≤ λz(x) ≤ 1 in Ω. u(x) being a minimizer for our functional, we have or Since A(x,ξ) is convex, the first term on the right-hand side can be evaluated in such a way that From (3.12) and (3.13), using Young's inequality, we obtain (3.18)

Let us evaluate now the term
So using Hypothesis 2.9, from which, choosing ε in a suitable way, we obtain Let us introduce now the function ϕ 1 ∈ C ∞ (Ω), defined by Let us put χ = 2pq/(q − 2p), and let us fix r > 0, s > m 1 . We define (3.25) Let us put now
We define the following function: From the definition of the function v(x) and (3.39), we have for all r > 0 and s > m 1 ( q/q). Now, we can organize the iterative Moser method (see [6]). We introduce for i = 0,1,2,...,   From this assertion, we deduce the inequality (3.4). Now, using [8,Lemma 4.8] and the interior regularity result of [5], we get the conclusion of Theorem 2.11.

Examples
Now, we describe a situation where hypotheses stated in Section 2 are satisfied.