1st group: Serena Benigno, Itahisa Barrios Cubas, Ricardo Ziegele
Speaker: Serena Benigno (Università della Campania)
Title: The Spectral Drop Problem in a bounded domain with Mixed Boundary Conditions
Abstract: We study the Spectral Drop Problem in a bounded domain in the case of mixed Dirichlet–Neumann boundary conditions. The domain is assumed to be sufficiently regular, and its boundary is decomposed into two smooth portions where the two types of conditions are prescribed, meeting along a smooth interface. Depending on the dimension, these portions are either submanifolds of codimension one or simply the endpoints of an interval. The problem consists in minimizing the first eigenvalue of the Laplacian over all admissible subsets of the domain having fixed measure. The admissible sets are taken in the class of quasi-open sets, and the associated eigenvalue is defined through a variational characterization involving functions that vanish on the Dirichlet part of the boundary and quasi-everywhere outside the chosen subset. We first show that the eigenvalue associated with a given subset always admits a minimizer, represented by a nonnegative eigenfunction. We then prove that the problem of the minimization of the eigenvalue among all the admissible sets which satisfy a measure constraint admits a solution, and that the volume constraint is saturated. Moreover, we study the regularity properties of the corresponding eigenfunction. In addition, we analyze qualitative features of optimal sets in specific geometries. In such cases, the interplay between the imposed boundary conditions and the geometry of the domain determines both the location and the shape of the optimal configuration. Finally, we examine the behavior of optimal sets as their volume tends to zero, focusing on the concentration phenomena that arise in this asymptotic regime. This is based on a joint work with Benedetta Pellacci (University of Campania “Luigi Vanvitelli”) and Hugo Tavares (Instituto Superior Tecnico-Universidade de Lisboa).
Speaker: Itahisa Barrios Cubas (Universidad Autónoma de Madrid)
Title: Pohozaev Identity for the Spectral Fractional Laplacian
Abstract: In this poster, we present a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to prove non-existence results for certain semilinear problems. The first work on nonlocal Pohozaev identities goes back to the influential paper by Ros-Oton and Serra (2014), where they consider the Restricted Fractional Laplacian (RFL) on \mathbb{R}^N. However, in our setting, the boundary behavior is significantly different, and their integration by parts approach cannot be applied. Instead, we adopt a spectral approach to derive a new identity expressed as a Schur product, through which we recover the classical Pohozaev identity for the Laplacian via a transition matrix.
This is joint work with Matteo Bonforte, María del Mar González, and Clara Torres-Latorre.
Speaker: Ricardo Ziegele (Università di Torino)
Title: Nodal solutions for the Minkowski mean curvature operator: Multiplicity and large-$\mu$ asymptotics
Abstract: We study the Dirichlet problem for the mean curvature operator in Minkowski space, on a bounded domain \(\Omega \subset \mathbb{R}^n\). For this problem, we study the interaction between two parameters \((\lambda,\mu)\) and their effects on the multiplicity and general behaviour of nodal solutions. In the general setting, using non-smooth variational methods, we prove the existence of a ground-state solution and a linking solution. In the radial case, we prove the existence of an arbitrarily large number of nodal solutions, up to taking the \(\mu\) large enough. Finally, we characterize the limiting profile of these solutions in both settings as \(\mu \to + \infty\). This is a joint work with Alberto Boscaggin (UNITO) and Francesca Colasuonno (UNITO).