EXTREMUM PRINCIPLE FOR VERY WEAK SOLUTIONS OF A-HARMONIC EQUATION

摘要

This paper deals with the very weak solutions of A-harmonic equation divA(x,u(x))=0 (*)where the operator A satisfies the monotonicity inequality, the controllable growth condition and the homogeneity condition. The extremum principle for very weak solutions of A-harmonic equation is derived by using the stability result of Iwaniec-Hodge decomposition: There exists an integrable exponent r1=r1(p,n,β/α)=1/2[p-α/100n2β+√(p+α/100n2β)2-4α/100n2βsuch that if u(x) ∈ W1,r(Ω) is a very weak solution of the A-harmonic equation(*),andm ≤ u(x)≤ M on ( )Ω in the Sobolev sense, then m ≤ u(x) ≤ M almost everywhere in Ω, provided that r ＞ r1. As a corollary, we prove that the 0-Dirichlet boundary value problem {divA(x, u(x)) =0u∈W01,r(Ω)of the A-harmonic equation has only zero solution if r＞r1.As a corollary,we prove that the 0-Dirichlet boundary value problem{divA(x,()u()x)=0;u ∈W1,r/0(Ω) of the A-harmonic equation has only zero solution if r>r1.