Geometric-type Sobolev inequalities and applications to the regularity of minimizers

Details Geometric-type Sobolev inequalities and applications to the regularity of minimizers Geometric-type Sobolev inequalities and applications to the regularity of minimizers

2011-11-11

arxiv.org/abs/1111.2801v1

Mathematics

Analysis of PDEs

20 pages; 1 figure

Scientific paper

The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-\Delta u=g(u)$ in a bounded domain $\Omega$ of $\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(\Omega)$ bound for the extremal solution of $-\Delta u=\lambda f(u)$ when $n\geq 5$ and the domain is convex. We recall that extremal solutions are known to be bounded in convex domains if $n\leq 4$, and that their boundedness is expected ---but still unkwown--- for $n\leq 9$.

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