arXiv subjects

Federico Villone

Publications and source records attributed to Federico Villone.

3 recordsLinked to original sources

arXiv2026

Symmetry-breaking and local stability of a two-phase eigenvalue problem in optimal insulation

We consider the first eigenvalue, $\lambda_\beta(\Omega,A)$, of a two-phase eigenvalue problem for the Laplacian with Robin boundary conditions, where the two phases are characterised by different ellipticity constants. We characterise the conditions under which a ball $B_R$ is a local minimum under a volume constraint for the minimisation problem $A\mapsto\lambda_\beta(B_r,A)$, in terms of the principal Neumann eigenvalue of the fixed inner ball $B_r$.

math.AP↗

arXiv2026

On some functionals involving torsional rigidity, principal eigenvalue and perimeter

In this paper we study some relationships between the first Dirichlet eigenvalue $\Lambda(\Omega)$ and the torsional rigidity $T(\Omega)$ of a domain $\Omega$. We consider the problem of optimizing the product $\Lambda(\Omega)T(\Omega)$ among sets with prescribed perimeter, both in the class of open sets with finite perimeter and within the class of convex domains. We also present local results for the quantity $\Lambda(\Omega)T(\Omega)^q$, with $q>0$, under either a volume or a perimeter constraint.

math.SP↗

arXiv2025

Remarks on the reinforcement of the spectrum of an elliptic problem with Robin boundary condition

We investigate the spectral properties of a differential elliptic operator on $H^1(\bar{\Omega}\cup \Sigma)$, where $\Omega$ is a smooth domain surrounded by a layer $\Sigma$. The thickness of the layer is given by $\varepsilon h$, where $h$ is a positive function defined on the boundary $\partial \Omega$ and $\varepsilon$ is the ellipticity constant of the operator in $\Sigma$. We prove that, in the limit for $\varepsilon$ going to $0$, the spectrum converges to the spectrum of a differential elliptic operator in $H^1(\Omega)$, and we investigate a first-order asymptotic development.

math.AP↗