On the behaviour of the solutions to p-Laplacian equations as p goes to 1

By Anna Mercaldo, Sergio Segura de León, Cristina Trombetti. 2008

56 citations recorded by the source corpus.

In the present paper we study the behaviour as $p$ goes to $1$ of the weak solutions to the problems $$ \begin{cases} -\operatorname{div} \bigl(|\nabla u_p|^{p-2}\nabla u_p\bigr)=f &\text{in } \Omega\\ u_p=0 &\text{on } \partial\Omega, \end{cases} $$ where $\Omega$ is a bounded open set of ${\mathbb R}^N$ $(N\ge 2)$ with Lipschitz boundary and $p > 1$. As far as the datum $f$ is concerned, we analyze several cases: the most general one is $f\in W^{-1,\infty}(\Omega)$. We also illustrate our results by means of remarks and examples.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/50fe6151db1b156a2fc200537c8dea0cf56332ea