A stability result for the first robin-neumann eigenvalue: a double perturbation approach

Let Formula: see text, Formula: see text, where Formula: see text and Formula: see text are two open, bounded and convex sets such that Formula: see text and let Formula: see text be a given parameter. We consider the eigenvalue problem for the Laplace operator associated to Formula: see text, with Robin boundary condition on Formula: see text and Neumann boundary condition on Formula: see text. In 47 it is proved that the spherical shell is the only maximizer for the first Robin–Neumann eigenvalue in the class of domains Formula: see text with fixed outer perimeter and volume. We establish a quantitative version of the afore-mentioned isoperimetric inequality; the main novelty consists in the introduction of a new type of hybrid asymmetry, that turns out to be the suitable one to treat the different conditions on the outer and internal boundary. Up to our knowledge, in this context, this is the first stability result in which both the outer and the inner boundary are perturbed.