TY - JOUR

T1 - Asymptotic spherical shapes in some spectral optimization problems

AU - Mazzoleni, Dario Cesare Severo

AU - Pellacci, Benedetta

AU - Verzini, Gianmaria

PY - 2019

Y1 - 2019

N2 - We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box Ω⊂RN, 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 led 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. Using α-symmetrization techniques on cones, we prove that, for suitable choices of the box Ω, the optimal shapes for this second problem are indeed spherical. Moreover, for general Ω, we show that small volume spectral drops are asymptotically spherical, centered near points of ∂Ω having largest mean curvature.

AB - We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box Ω⊂RN, 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 led 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. Using α-symmetrization techniques on cones, we prove that, for suitable choices of the box Ω, the optimal shapes for this second problem are indeed spherical. Moreover, for general Ω, we show that small volume spectral drops are asymptotically spherical, centered near points of ∂Ω having largest mean curvature.

KW - Mixed Neumann-Dirichlet boundary conditions

KW - Singular limits

KW - Survival threshold

KW - α-symmetrization

KW - Mixed Neumann-Dirichlet boundary conditions

KW - Singular limits

KW - Survival threshold

KW - α-symmetrization

UR - http://hdl.handle.net/10807/143892

UR - http://www.elsevier.com/locate/jmpa

U2 - 10.1016/j.matpur.2019.10.002

DO - 10.1016/j.matpur.2019.10.002

M3 - Article

SP - 256

EP - 283

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

ER -