On the Sets of Regularity of Solutions for a Class of Degenerate Nonlinear Elliptic Fourth-Order Equations with Data

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Research Article On the Sets of Regularity of Solutions for a Class of Degenerate Nonlinear Elliptic Fourth-Order Equations with L1 Data S. Bonafede and F. Nicolosi Received 24 January 2007; Accepted 29 January 2007 Recommended by V. Lakshmikantham

We establish H¨older continuity of generalized solutions of the Dirichlet problem, associated to a degenerate nonlinear fourth-order equation in an open bounded set Ω ⊂ Rn , with L1 data, on the subsets of Ω where the behavior of weights and of the data is regular enough. Copyright © 2007 S. Bonafede and F. Nicolosi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction ◦ 1,q

In this paper, we will deal with equations involving an operator A : W 2,p (ν,μ,Ω) → ◦ 1,q (W 2,p (ν,μ,Ω)) of the form 

Au =





(−1)|α| Dα Aα x, ∇2 u ,

(1.1)

|α|=1,2



where Ω is a bounded open set of Rn , n > 4, 2 < p < n/2, max(2p, n) < q < n, ν and μ are ◦ 1,q positive functions in Ω with properties precised later, W 2,p (ν,μ,Ω) is the Banach space of all functions u : Ω → R with the properties |u|q ,ν|Dα u|q ,μ|Dβ u| p ∈ L1 (Ω), |α| = 1, |β| = 2, and “zero” boundary values; ∇2 u = {Dα u : |α| ≤ 2}. The functions Aα satisfy growth and monotonicity conditions, and in particular, the following strengthened ellipticity condition (for a.e. x ∈ Ω and ξ = {ξα : |α| = 1,2}): 



Aα (x,ξ)ξα ≥ c2

|α|=1,2

where c2 > 0, g2 (x) ∈ L1 (Ω).

 |α|=1

 q   p ν(x)ξα  + μ(x)ξα  |α|=2

 − g2 (x),

(1.2)

2

Boundary Value Problems

We will assume that the right-hand sides of our equations, depending on unknown function, belong to L1 (Ω). A model representative of the given class of equations is the following: −





D

α

|α|=1

ν

   Dβ u2

(q−2)/2

α

D u +





D

α

|α|=2

|β|=1

μ

   D β u 2

(p−2)/2

α

D u

|β|=2

= −|u|σ −1 u + f

in Ω, (1.3)

where σ > 1 and f ∈ L1 (Ω). The assumed conditions and known results of the theory of monotone operators allow us to prove existence of generalized solutions of the Dirichlet problem associated to our operator (see, e.g., [1]), bounded on the sets G ⊂ Ω where the behavior of weights and of the data of the problem is regular enough (see [2]). In our paper, following the approach of [3], we establish on such sets a result on H¨older continuity of generalized solutions of the same Dirichlet problem. We note that for one high-order equation with degenerate nonlinear operator satisfying a strengthened ellipticity condition, regularity of solutions was studied in [4, 5] (nondegenerate case) and in [6, 7] (degenerate case). However, it has been made for equations with right-hand sides in Lt with t > 1. 2. Hypotheses Let n ∈ N, n > 4, and let Ω be a bounded open set of Rn . Let p, q be two real numbers √ such that 2 < p < n/2, max(2p, n) < q < n. Let ν : Ω → R+ be a measurable function such that 1/(q−1)

1 ν

ν ∈ L1loc (Ω),

∈ L1loc (Ω).

(2.1)

W 1,q

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