## Differential and Integral Equations

### Anisotropic equations: Uniqueness and existence results

#### 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