## D. Mazzoleni - B. Pellacci - G. Verzini

# Asymptotic spherical shapes in some spectral optimization problems

created by mazzoleni on 06 Nov 2018

modified on 21 Nov 2018

[

BibTeX]

*Preprint*

**Inserted:** 6 nov 2018

**Last Updated:** 21 nov 2018

**Year:** 2018

**Abstract:**

We study the optimization of the positive principal eigenvalue of an
indefinite weighted problem,
associated with the Neumann Laplacian in a box $\Omega\subset\mathbb{R}^N$, which arises in
the investigation of the survival threshold in population dynamics.
When trying to minimize such eigenvalue with respect to the weight, one is lead to
consider a shape optimization problem, which is known to admit no spherical optimal
shapes (despite some previously stated conjectures).
We investigate whether spherical shapes can be recovered in some singular perturbation
limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit
the so called spectral drop problem, which involves the minimization of the first
eigenvalue of the mixed Dirichlet-Neumann Laplacian. We prove that, for suitable choices
of the box $\Omega$, the optimal shapes for this second problem are indeed spherical;
moreover, for general $\Omega$, we show that small volume spectral drops are
asymptotically spherical, with center at points of $\partial\Omega$ having large mean curvature.

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