Optimal Szegö-Weinberger type inequalities

Gabriella Di Blasio, Francesco Chiacchio, Friedemann Brock
2016 Communications on Pure and Applied Analysis  
Denote with µ 1 (Ω; e h(|x|) ) the first nontrivial eigenvalue of the Neumann problem where Ω is a bounded and Lipschitz domain in R N . Under suitable assumption on h we prove that the ball centered at the origin is the unique set maximizing µ 1 (Ω; e h(|x|) ) among all Lipschitz bounded domains Ω of R N of prescribed e h(|x|) dx-measure and symmetric about the origin. Moreover, an example in the model case h (|x|) = |x| 2 , shows that, in general, the assumption on the symmetry of the domain
more » ... annot be dropped. In the one-dimensional case, i.e. when Ω reduces to an interval (a, b), we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval (a, b) slides along the x-axis keeping fixed its weighted length.
doi:10.3934/cpaa.2016.15.367 fatcat:z6jcq2watzakfb5dx7ozsh4lxi