Estimates for the Sobolev trace constant with critical exponent and applications

Details Estimates for the Sobolev trace constant with critical exponent and applications Estimates for the Sobolev trace constant with critical exponent and applications

2006-12-13

arxiv.org/abs/math/0612354v1

Ann. Mat. Pura Appl, 187 (2008), no. 4, 683--704.

Mathematics

Analysis of PDEs

22 pages, submitted

Scientific paper

10.1007/s10231-007-0062-1

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $S\|u\|^p_{L^{p_*}(\partial\Omega) \hookrightarrow \|u\|^p_{W^{1,p}(\Omega)}$ that are independent of $\Omega$. This estimates generalized those of [3] for general $p$. Here $p_* := p(N-1)/(N-p)$ is the critical exponent for the immersion and $N$ is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of [16]. Finally, we study an optimal design problem with critical exponent.

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