Differential and Integral Equations

Anisotropic equations: Uniqueness and existence results

Stanislav Antontsev and Michel Chipot

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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.

Article information

Source
Differential Integral Equations, Volume 21, Number 5-6 (2008), 401-419.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356038624

Mathematical Reviews number (MathSciNet)
MR2483261

Zentralblatt MATH identifier
1224.35088

Subjects
Primary: 35K65: Degenerate parabolic equations
Secondary: 35J70: Degenerate elliptic equations 35K60: Nonlinear initial value problems for linear parabolic equations 35K65: Degenerate parabolic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Citation

Antontsev, Stanislav; Chipot, Michel. Anisotropic equations: Uniqueness and existence results. Differential Integral Equations 21 (2008), no. 5-6, 401--419. https://projecteuclid.org/euclid.die/1356038624


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