# Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

##
Daniele Castorina
(Università di Napoli "Federico II")

created by novaga on 10 Jan 2017

1 feb 2017
-- 17:00
[open in google calendar]

Aula Seminari, Dipartimento di Matematica di Pisa

**Abstract.**

We prove a Sobolev and a Morrey type inequality involving the mean
curvature and the tangential gradient with respect to the level sets
of the function that appears in the inequalities. Then, as an
application, we establish \textit{a priori} estimates for semistable
solutions of $-\Delta_p u= g(u)$ in a smooth bounded domain
$\Omega\subset \mathbb{R}^n$. In particular, we obtain new
$L^r$ and $W^{1,r}$ bounds for the extremal solution
$u^\star$ when the domain is strictly convex. More precisely, we prove that
$u^\star\in L^\infty(\Omega)$ if $n\leq p+2$ and $u^\star\in
L^{\frac{np}{n-p-2}}(\Omega)\cap W^{1,p}_0(\Omega)$ if $n>p+2$.
This is a joint work with Manel Sanchon (UAB Barcelona).