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2008 Anisotropic equations: Uniqueness and existence results
Stanislav Antontsev, Michel Chipot
Differential Integral Equations 21(5-6): 401-419 (2008).

Abstract

We study uniqueness of weak solutions for elliptic equations of the following type \[ -\partial_{x_{i}}\left( a_{i}(x,u)\left\vert \partial_{x_{i}}u\right\vert ^{p_{i}-2}\partial_{x_{i}}u\right) +b(x,u) =f(x), \] in a bounded domain $\Omega\subset{\mathbb{R}}^{n}$ with Lipschitz continuous boundary $\Gamma=\partial\Omega$. We consider in particular mixed boundary conditions, i.e., Dirichlet condition on one part of the boundary and Neumann condition on the other part. We study also uniqueness of weak solutions for the parabolic equations \[ \left\{ \begin{array}[c]{cc} \partial_{t}u=\partial_{x_{i}}\left( a_{i}(x,t,u)\left\vert \partial_{x_{i} }u\right\vert ^{p_{i}-2}\partial_{x_{i}}u\right) +f & \text{in}\ \Omega \times(0,T),\\ u=0 & \text{on\ }\Gamma\times(0,T)=\partial\Omega\times(0,T),\\ u(x,0)=u_{0} & x\in\Omega. \end{array} \right. \] It is assumed that the constant exponents $p_{i}$ satisfy $1 <p_{i} <\infty$ and the coefficients $a_{i}\,$ are such that $0 <\lambda\leq\lambda_{i}\leq a_{i}(x,u) <\infty,\ \forall i,a.e. \; x\in\Omega$, (a.e. $t\in(0,T)$), $\forall u\in{\mathbb{R}}$. We indicate also conditions which guarantee existence of solutions.

Citation

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Stanislav Antontsev. Michel Chipot. "Anisotropic equations: Uniqueness and existence results." Differential Integral Equations 21 (5-6) 401 - 419, 2008.

Information

Published: 2008
First available in Project Euclid: 20 December 2012

zbMATH: 1224.35088
MathSciNet: MR2483261

Subjects:
Primary: 35K65
Secondary: 35J70, 35K60, 35K65, 46E35

Rights: Copyright © 2008 Khayyam Publishing, Inc.

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Vol.21 • No. 5-6 • 2008
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