We consider the constrained critical values of the Dirichlet integral on the unit Lq sphere. This leads to a semilinear eigenvalue–type equation. We focus on some properties of the relevant spectrum, such as discreteness and simplicity and isolation of the first eigenvalue. We also consider the problem of providing sharp geometric estimates of the first eigenvalue, in terms of volume, inradius and isoperimetric ratios.
A list of open problems will be included.
We study the positive principal eigenvalue of a weighted problem associated with the Neumann-Laplacian settled in a box Ω⊂RN, which arises from the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the sign-changing weight, one is lead to consider a shape optimization problem, which is known to admit spherical optimal shapes only in trivial cases. We investigate if spherical shapes can be recovered in the limit when the negative part of the weight diverges. First of all, we show that the shape optimization problem appearing in the limit is the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Thanks to α-symmetrization techniques on cones, it can be proved that optimal shapes for the spectral drop problem are spherical for suitable choices of the box, the most interesting case being when Ω is a convex polytope, and in this case a quantitative analysis of the convergence can be performed. Finally, for a smooth Ω, we show that small volume spectral drops are asymptotically spherical, with center at points with high mean curvature.
This is a joint project with Benedetta Pellacci and Gianmaria Verzini.
In this talk we study the asymptotic behavior of u-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets Ω and ω of R2, containing the origin. First, if ε is positive and small enough and if u is a function defined on Ω, we compute an asymptotic expansion of the u-capacity CapΩ(εω,u) as ε→0. Then, by exploiting the expression for CapΩ(εω,u), under suitable assumptions, we compute an asymptotic expansion for the N-th eigenvalues of the Dirichlet-Laplacian in the perforated set Ω\(εω) for ε close to 0. Such formula shows explicitly the dependence of the asymptotic expansion on the behavior of the corresponding eigenfunction near 0 and on the shape ω of the hole.
Based on joint work with L. Abatangelo, V. Bonnaillie-Noël, and C. Léna.