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
Stanislav Antontsev. Michel Chipot. "Anisotropic equations: Uniqueness and existence results." Differential Integral Equations 21 (5-6) 401 - 419, 2008. https://doi.org/10.57262/die/1356038624
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