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Volume 18, Issue 3
Extremum Principle for Very Weak Solutions of A-harmonic Equation

Hongya Gao , Juan Li & Yanjun Deng

J. Part. Diff. Eq., 18 (2005), pp. 235-240.

Published online: 2005-08

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  • 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.

  • Keywords

A-harmonic equation extremum principle very weak solution Iwaniec-Hodge decomposition

  • AMS Subject Headings

35J60 35J67

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-18-235, author = {Hongya Gao , Juan Li and Yanjun Deng }, title = {Extremum Principle for Very Weak Solutions of A-harmonic Equation}, journal = {Journal of Partial Differential Equations}, year = {2005}, volume = {18}, number = {3}, pages = {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.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5359.html} }
TY - JOUR T1 - Extremum Principle for Very Weak Solutions of A-harmonic Equation AU - Hongya Gao , Juan Li & Yanjun Deng JO - Journal of Partial Differential Equations VL - 3 SP - 235 EP - 240 PY - 2005 DA - 2005/08 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5359.html KW - A-harmonic equation KW - extremum principle KW - very weak solution KW - Iwaniec-Hodge decomposition AB -

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.

Hongya Gao , Juan Li and Yanjun Deng . (2005). Extremum Principle for Very Weak Solutions of A-harmonic Equation. Journal of Partial Differential Equations. 18 (3). 235-240. doi:
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