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Pier Domenico Lamberti

Publications and source records attributed to Pier Domenico Lamberti.

At least 19 recordsLinked to original sources
arXiv2026

On a shape optimisation problem for Maxwell's eigenvalues on cuboids

We consider an optimisation problem for the elementary symmetric functions of the first three Maxwell's eigenvalues on cuboids under volume and perimeter constraint, and we show that the cube is a local minimiser. More precisely, it is the unique minimiser in an explicit cone of cuboids. The result gives a model case for the local optimisation of Maxwell's eigenvalues. On the other hand we show that such local extremality phenomena cannot be expected outside rigid geometric classes.

math.SP↗
arXiv2026

Asymptotic behavior and spectral distortion for biharmonic Steklov problems on thin domains

In this paper, we investigate the asymptotic behavior of the eigenvalues and eigenfunctions of a biharmonic Steklov problem defined on a thin domain in the $n$ dimensional Euclidean space degenerating to a segment. For $n=2$ the problem models the vibrations of a thin elastic plate with cross section represented by the given domain and mass concentrated on a free boundary. The problem under consideration depends on a parameter $σ$ that in the theory of elastic plates represents the Poisson ratio of the material. Our analysis points out a distortion in the limiting problem depending on $σ$ and the space dimension $n$.

math.AP↗
arXiv2026

The nonlinear Steklov problem in outward cuspidal domains

In this article, we consider the nonlinear Steklov eigenvalue problem in outward cuspidal domains. Using the compactness of the weighted trace embedding we obtain the variational characterization of the first non-trivial eigenvalue and prove the existence of a corresponding weak solution.

math.AP↗
arXiv2025

Steklov vs. Steklov: A Fourth-Order Affair Related to the Babuška Paradox

We discuss two fourth-order Steklov problems and highlight a Babuška paradox appearing in their approximations on convex domains via sequences of convex polygons. To do so, we prove that the eigenvalues of one of the two problems depend with continuity upon domain perturbation in the class of convex domains, extending a result known in the literature for the first eigenvalue. This is obtained by examining in detail a nonlocal, second-order problem for harmonic functions introduced by Ferrero, Gazzola, and Weth. We further review how this result is connected to diverse variants of the classical Babuška paradox for the hinged plate and to a degeneration result by Maz'ya and Nazarov.

math.AP↗
arXiv2025

Shape sensitivity analysis of Neumann-Poincaré eigenvalues

This paper concerns the eigenvalues of the Neumann-Poincaré operator, a boundary integral operator associated with the harmonic double-layer potential. Specifically, we examine how the eigenvalues depend on the support of integration and prove that the map associating the support's shape to the eigenvalues is real-analytic. We then compute its first derivative and present applications of the resulting formula. The proposed method allows for handling infinite-dimensional perturbation parameters for multiple eigenvalues and perturbations that are not necessarily in the normal direction.

math.AP↗
arXiv2025

Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part II

In this second part of our series of papers, we develop an abstract framework suitable for de Rham complexes that depend on a parameter belonging to an arbitrary Banach space. Our primary focus is on spectral perturbation problems and the differentiability of eigenvalues with respect to perturbations of the involved parameters. As a byproduct, we provide a proof of the celebrated Hellmann-Feynman theorem for both simple and multiple eigenvalues of suitable families of self-adjoint operators in Hilbert spaces, even when these operators depend on possibly infinite-dimensional parameters. We then apply this abstract machinery to the de Rham complex in three dimensions, considering mixed boundary conditions and non-constant coefficients. In particular, we derive Hadamard-type formulas for Maxwell and Helmholtz eigenvalues. First, we compute the derivatives under minimal regularity assumptions - specifically, Lipschitz regularity - on both the domain and the perturbation, expressing the results in terms of volume integrals. Second, under more regularity assumptions on the domains, we reformulate these formulas in terms of surface integrals.

math.SP↗
arXiv2025

Shape Derivatives of the Eigenvalues of the De Rham Complex for Lipschitz Deformations and Variable Coefficients: Part I

We study eigenvalue problems for the de Rham complex on varying three dimensional domains. Our analysis includes the Helmholtz equation as well as the Maxwell system with mixed boundary conditions and non-constant coefficients. We provide Hadamard-type formulas for the shape derivatives under weak regularity assumptions on the domain and its perturbations. Our proofs are based on abstract results adapted to varying Hilbert complexes. As a bypass product of our analysis we give a proof of the celebrated Helmann-Feynman theorem both for simple and multiple eigenvalues of suitable families of self-adjoint operators in Hilbert space depending on possibly infinite dimensional parameters. This series of papers consists of Parts I and II.

math.AP↗
arXiv2024

On the Phragmén-Lindelöf and the superposition principles for the $p$-Laplacian

We study sub and supersolutions for the $p$-Laplace type elliptic equation of the form $$-Δ_p u-V|u|^{p-2}u=0\quad\text{in $Ω$},$$ where $Ω$ is a radially symmetric domain in ${\mathbb{R}}^N$ and $V(x)\ge 0$ is a continuous potential such that the solutions of the equation satisfy the comparison principle on bounded subdomains of $Ω$. In this work we establish a superposition principle and then use it to develop a version of a Phragmén-Lindelöf comparison principle in the case $p\ge 2$. Moreover, by applying this principle to the case of Hardy-type potentials we recover and improve a number of known lower and upper estimates for sub and supersolutions.

math.AP↗
arXiv2023

Permittivity optimization for Maxwell's eigenvalues

We formulate an optimization problem for the dependence of the eigenvalues of Maxwell's equations in a cavity upon variation of the electric permittivity and we prove a corresponding Maximum Principle.

math.AP↗
arXiv2023

Extension and embedding theorems for Campanato spaces on $C^{0,γ}$ domains

We consider Campanato spaces with exponents $λ, p$ on domains of class $C^{0,γ}$ in the N-dimensional Euclidean space endowed with a natural anisotropic metric depending on $γ$. We discuss several results including the appropriate Campanato's embedding theorem and we prove that functions of those spaces can be extended to the whole of the Euclidean space without deterioration of the exponents $λ, p$.

math.FA↗
arXiv2022

On a Steklov spectrum in Electromagnetics

After presenting various concepts and results concerning the classical Steklov eigenproblem, we focus on analogous problems for time-harmonic Maxwell's equations in a cavity. In this direction, we discuss recent rigorous results concerning natural Steklov boundary value problems for the curlcurl operator. Moreover, we explicitly compute eigenvalues and eigenfunctions in the unit ball of the three-dimensional Euclidean space by using classical vector spherical harmonics.

math.AP↗
arXiv2022

Spectral stability of the $curl curl$ operator via uniform Gaffney inequalities on perturbed electromagnetic cavities

We prove spectral stability results for the $curl curl$ operator subject to electric boundary conditions on a cavity upon boundary perturbations. The cavities are assumed to be sufficiently smooth but we impose weak restrictions on the strength of the perturbations. The methods are of variational type and are based on two main ingredients: the construction of suitable Piola-type transformations between domains and the proof of uniform Gaffney inequalities obtained by means of uniform a priori $H^2$-estimates for the Poisson problem of the Dirichlet Laplacian. The uniform a priori estimates are proved by using the results of V. Maz'ya and T. Shaposhnikova based on Sobolev multipliers. Connections to boundary homogenization problems are also indicated.

math.AP↗
arXiv2021

Spectral stability of the Steklov problem

This paper investigates the stability properties of the spectrum of the classical Steklov problem under domain perturbation. We find conditions which guarantee the spectral stability and we show their optimality. We emphasize the fact that our spectral stability results also involve convergence of eigenfunctions in a suitable sense according with the definition of connecting system by \cite{Vainikko}. The convergence of eigenfunctions can be expressed in terms of the $H^1$ strong convergence. The arguments used in our proofs are based on an appropriate definition of compact convergence of the resolvent operators associated with the Steklov problems on varying domains. In order to show the optimality of our conditions we present alternative assumptions which give rise to a degeneration of the spectrum or to a discontinuity of the spectrum in the sense that the eigenvalues converge to the eigenvalues of a limit problem which does not coincide with the Steklov problem on the limiting domain.

math.AP↗
arXiv2021

Spectral stability for a class of fourth order Steklov problems under domain perturbations

We study the spectral stability of two fourth order Steklov problems upon domain perturbation. One of the two problems is the classical DBS - Dirichlet Biharmonic Steklov - problem, the other one is a variant. Under a comparatively weak condition on the convergence of the domains, we prove the stability of the resolvent operators for both problems, which implies the stability of eigenvalues and eigenfunctions. The stability estimates for the eigenfunctions are expressed in terms of the strong $H^2$-norms. The analysis is carried out without assuming that the domains are star-shaped. Our condition turns out to be sharp at least for the variant of the DBS problem. In the case of the DBS problem, we prove stability of a suitable Dirichlet-to-Neumann type map under very weak conditions on the convergence of the domains and we formulate an open problem. As bypass product of our analysis, we provide some stability and instability results for Navier and Navier-type boundary value problems for the biharmonic operator.

math.AP↗
arXiv2020

Shape perturbation of Grushin eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $\mathbb{R}^N$. We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich-Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich-Pohozaev identity for the Grushin eigenvalues.

math.AP↗