Calculus of Variations and Geometric Measure Theory

G. Palatucci

Subcritical approximation of the Sobolev quotient and a related concentration result

created by palatucci on 17 Feb 2011
modified on 23 Dec 2011


Published Paper

Inserted: 17 feb 2011
Last Updated: 23 dec 2011

Journal: Rend. Sem. Mat. Univ. Padova
Volume: 125
Pages: 1-14
Year: 2011


Let $\Omega$ be a general, possibly non-smooth, bounded domain of $\mathbb{R}^N$, $N\geq 3$. Let $\displaystyle 2^{*}\!\!=\!{2N}\,\!/{(N-2)}$ be the critical Sobolev exponent. We study the following variational problem $$ S{}{\varepsilon}=\sup\left \{ \int{\Omega}
}\!-\varepsilon}dx: \int{\Omega}
{2}dx\leq 1, u=0 \ \text{on} \ \partial\Omega \right \}, $$ investigating its asymptotic behavior as $\varepsilon$ goes to zero, by means of $\Gamma^{+}$-convergence techniques. We also show that sequences of maximizers $u_\varepsilon$ concentrate energy at one point $x_0\in \overline{\Omega}$.

Keywords: G-convergence, concentration, critical exponent, Sobolev inequality