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Seminario del 2022
2022-06-07
Dario Mazzoleni
Singular analysis of the optimizers of the principal eigenvalue in weighted Neumann problems
Seminario di analisi matematica
We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain \Omega\subset R^N, within a suitable class of sign-changing weights. This problem naturally arises in population dynamics.
Denoting with u the optimal eigenfunction and with
D its super-level set associated to the optimal weight,
we perform the analysis of the singular limit of the optimal eigenvalue
as the measure of D tends to zero.
We show that, when the measure of D is sufficiently small, u has a unique
local maximum point lying on the boundary of \Omega and D is connected.
Furthermore, the boundary of D intersects the boundary of the box \Omega, and more precisely, ${\mathcal H}^{N-1}(\partial D \cap \partial \Omega)\ge C|D|^{(N-1)/N}
$ for some universal constant C>0.
Though widely expected, these properties are still unknown if the measure of D is arbitrary.
This is a joint project with B. Pellacci and G. Verzini.