Consider the Poisson equation with Robin boundary conditions in a (nonsmooth) domain with a bounded, nonnegative right hand side. Given a point on the boundary, the question is whether the solution has a strictly positive lower limit at this point. If the domain is smooth the answer is positive as a consequence of the Hopf maximum principle. If the domain is not smooth, the answer may be positive or negative, depending on the geometry of the domain around the point. This question was raised in a probabilistic context by Bass, Burdzy and Chen in 2008, when they obtained results for Lipschitz sets and cuspidal domains.

Our motivation is related to the fact that positive answers to the question above, together with a control of the infimum of the boundary values, lead to sharp quantitative forms of isoperimetric inequalities of spectral type for the Robin Laplacian.

In this talk, I will make the point on recent advances on isoperimetric inequalities involving the Robin Laplacian and I will show how the boundary behaviour pops up in the quantitative form of the inequalities. If times remains, I will present a variational approach to deal with the boundary behaviour for general elliptic operators and various geometric contexts.

The results were obtained together with A. Giacomini and M. Nahon.