Year: 2005
Journal of Partial Differential Equations, Vol. 18 (2005), Iss. 3 : pp. 235–240
Abstract
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 r_1 = r_1\left(p, n, \frac{β}{α}\right) = \frac{1}{2} \bigg[p - \frac{α}{100n²β} + \sqrt{\left(p + \frac{α}{100n²β}\right)² - \frac{4α}{100n²β}}\bigg] 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 ∂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)) = 0 u ∈ W^{1, r}_0 (Ω) of the A-harmonic equation has only zero solution if r > r_1.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2005-JPDE-5359
Journal of Partial Differential Equations, Vol. 18 (2005), Iss. 3 : pp. 235–240
Published online: 2005-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 6
Keywords: A-harmonic equation