EXTREMUM PRINCIPLE FOR VERY WEAK SOLUTIONS OF A-HARMONIC EQUATION

This paper deals with the very weak solutions of A-harmonic equation divA(x, vu(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 such that if u(x) is an element of W-l,W-r (Omega) is a very weak solution of the A-harmonic equation (*), and m <= u(x) <= M on partial derivative Omega in the Sobolev sense, then m <= u(x) <= M almost everywhere in Omega, provided that r > r(1). As a corollary, we prove that the 0-Dirichlet boundary value problem {divA(x, del u(x)) = 0 u is an element of W-0(1'r) (Omega) of the A-harmonic equation has only zero solution if r > r(1).