On the behaviour of the first eigenvalue of the $p$-Laplacian with Robin boundary conditions as $p$ goes to $1$

In this paper we study the $\mathrm{\Gamma }$-limit, as $p\to 1$, of the functional $$J_p(u)=\frac{{\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+\beta {\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^p}}{{\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^p}},$$ where $\mathrm{\Omega }$ is a smooth bounded open set in ${\mathbb{R}}^N$, $p>1$ and $\beta$ is a real number. Among our results, for $\beta >-1$, we derive an isoperimetric inequality for $$\mathrm{\Lambda }(\mathrm{\Omega },\beta )=\underset{u\in BV(\mathrm{\Omega }),u\equiv ̸0}{\inf }\frac{{\displaystyle \mathrm{\mid }Du\mathrm{\mid }(\mathrm{\Omega })+\min (\beta ,1){\int }_{\mathrm{\partial }\mathrm{\Omega }}\mathrm{\mid }u\mathrm{\mid }}}{{\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{\mid }u\mathrm{\mid }}}$$ which is the limit as $p\to 1^{+}$ of $ \lambda(\Omega,p,\beta)= \displaystyle \min_{u\in W^{1,p}(\Omega)} J_{p}(u). $ We show that among all bounded and smooth open sets with given volume, the ball maximizes $\mathrm{\Lambda }(\mathrm{\Omega },\beta )$ when $\beta \in$ $(-1,0)$ and minimizes $\mathrm{\Lambda }(\mathrm{\Omega },\beta )$ when $\beta \in [0,\mathrm{\infty })$.