Friday, December 2, 2022 — 12:15 pm — aula M2.3, edificio Matematica
Some shape optimization problems involving principal frequencies of the $p$-Laplacian
Abstract: In this talk we present some results concerning the optimisation problems for the scaling free functional
$$F_{p,\beta} (\Omega)=\frac{\lambda_p(\Omega)T_p^\beta(\Omega)}{|\Omega|^{\alpha(p,\beta,N)}}$$
and the limit shape functional
$$F_{\infty,\beta}(\Omega)= \lim_{p\to \infty}F^{1/p}_{p,\beta}(\Omega)$$
over the class of open sets $\Omega \subset \mathbb R^N$ with $0<|\Omega|<\infty$ and in that of bounded convex open sets. Here $1<p<\infty$, $\beta>0$, $\lambda_p(\Omega)$ is the principal eigenvalue relative to the $p$-Laplace operator, $T_p(\Omega)$ is the $p$-torsional rigidity and $\alpha(p,\beta,N):=\beta(p-1)+\frac{p(\beta-1)}{N}$. The study of the functionals $F_{p,\beta}$ has been already considered in the literature in case when $p=2, \beta=1$. The last part of the talk is devoted to giving some results dealing with the minimisation and maximisation problems of the functional
$$J_{p,q}(\Omega)=\frac{\lambda_p^{1/p}(\Omega)}{\lambda_q^{1/q}(\Omega)}.$$
All the results presented here are obtained in collaboration with G. Buttazzo (Pisa) and L. Briani (Pisa).