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 conditio...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/100n^2β＋√（p＋α/100n^2β）^2-4α/100n^2β] such that if u（x） ∈ W^1,r（Ω）is a very weak solution of the A-harmonic equation （＊）, and m ≤ u（x） ≤ M on ЭΩ in the Sobolev sense, then m ≤u（x） 〈 M almost everywhere in Ω, provided that r 〉 r1. As a corollary, we prove that the O-Dirichlet boundary value problem {div_A（x, u（x））=0,u∈W0^1,r（Ω）of the A-harmonic equation has only zero solution if r 〉 r1.展开更多
文摘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/100n^2β＋√（p＋α/100n^2β）^2-4α/100n^2β] such that if u（x） ∈ W^1,r（Ω）is a very weak solution of the A-harmonic equation （＊）, and m ≤ u（x） ≤ M on ЭΩ in the Sobolev sense, then m ≤u（x） 〈 M almost everywhere in Ω, provided that r 〉 r1. As a corollary, we prove that the O-Dirichlet boundary value problem {div_A（x, u（x））=0,u∈W0^1,r（Ω）of the A-harmonic equation has only zero solution if r 〉 r1.
