Differential and Integral Equations

Anisotropic equations: Uniqueness and existence results

Stanislav Antontsev and Michel Chipot

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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

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

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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