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

Castorina, Daniele, and Sanchón, Manel. "Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities." Journal of the European Mathematical Society 017.11 (2015): 2949-2975. <http://eudml.org/doc/277353>.

@article{Castorina2015,
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 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 $n\le p+2$ and $u^\star \in L^\{\frac\{np\}\{n-p-2\}\} (\Omega ) \cap W^\{1,p\}_0 (\Omega )$ if $n > p + 2$.},
author = {Castorina, Daniele, Sanchón, Manel},
journal = {Journal of the European Mathematical Society},
keywords = {geometric inequalities; mean curvature of level sets; Schwarz symmetrization; $p$-Laplace equations; regularity of stable solutions; geometric inequalities; mean curvature of level sets; Schwarz symmetrization; -Laplace equations; regularity of stable solutions},
language = {eng},
number = {11},
pages = {2949-2975},
publisher = {European Mathematical Society Publishing House},
title = {Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities},
url = {http://eudml.org/doc/277353},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Castorina, Daniele
AU - Sanchón, Manel
TI - Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 11
SP - 2949
EP - 2975
AB - 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 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 $n\le p+2$ and $u^\star \in L^{\frac{np}{n-p-2}} (\Omega ) \cap W^{1,p}_0 (\Omega )$ if $n > p + 2$.
LA - eng
KW - geometric inequalities; mean curvature of level sets; Schwarz symmetrization; $p$-Laplace equations; regularity of stable solutions; geometric inequalities; mean curvature of level sets; Schwarz symmetrization; -Laplace equations; regularity of stable solutions
UR - http://eudml.org/doc/277353
ER -