Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition

Shuhong Chen,Zhong Tan

doi:10.1155/2013/676215

Abstract

We consider boundary regularity for weak solutions of second-order quasilinear elliptic systems under natural growth condition with super quadratic growth and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

Highlights

This paper considers boundary regularity for weak solutions of quasilinear elliptic systems
Where Ω is a bounded domain in Rn with boundary of class C1, n ≥ 2 and u takes value in RN, N > 1
We assume that weak solutions exist and consider partial regularity of weak solutions directly

Summary

Introduction

This paper considers boundary regularity for weak solutions of quasilinear elliptic systems. Inspired by [14], in this paper, we would establish boundary regularity for quasilinear systems under natural growth condition by the method of A-harmonic approximation. For ] ∈ L1(Dρ(x0)), x0 ∈ D, we write ]󸀠x0,ρ = ∫− Dρ (x0 ) ]dHn−1 Combining this result with the analogous interior [19] and a standard covering argument allows us to obtain the following bound on the size of the singular set. Consider a bounded weak solution of (1) on the upper half unit ball B+ which satisfies the boundary condition (H5)󸀠 and ‖u‖L∞ ≤ M < ∞ with 2a(M)M < λ, where the structure conditions (H1)–(H3) hold for Aαijβ and (H4) holds for Bi. there exist positive R0 and ε0 (depending only on n, N, λ, L, b, M, a(M), M, ω(⋅), m, and γ) with the property that.

The A-Harmonic Approximation Technique

The Caccioppoli Inequality

The Proof of the Main Theorem