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Laura Abatangelo

Publications and source records attributed to Laura Abatangelo.

At least 19 recordsLinked to original sources
arXiv2026

Qualitative properties of eigenfunctions in domains with small holes

In this paper we study qualitative properties of the eigenvalues and eigenfunctions of $-\Delta$ with Dirichlet boundary condition in a smooth bounded domain $\Omega$ with a small circular hole. In the literature, this is known as a "singular perturbation", in contrast with the "regular perturbation" case. Denoting by $\Omega_\epsilon:=\Omega\setminus B(P,\epsilon)$ where $B(P,\epsilon)$ is the ball centered at $P$ and radius $\epsilon$, for $P\in\Omega$ and $\epsilon$ small enough we investigate 1) quantitative estimates for the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$; 2) the simplicity of the eigenvalues of $-\Delta$ in $\Omega_\epsilon$; 3) the behavior of nodal sets of the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$. A key ingredient in our analysis consists of pointwise estimates on the so-called $u$-capacitary potential firstly introduced in \cite{afhl}.

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arXiv2024

Bifurcation of double eigenvalues for Aharonov-Bohm operators with a moving pole

We study double eigenvalues of Aharonov-Bohm operators with Dirichlet boundary conditions in planar domains containing the origin. We focus on the behavior of double eigenvalues when the potential's circulation is a fixed half-integer number and the operator's pole is moving on straight lines in a neighborhood of the origin. We prove that bifurcation occurs if the pole is moving along straight lines in a certain number of cones with positive measure. More precise information is given for symmetric domains; in particular, in the special case of the disk, any eigenvalue is double if the pole is located at the centre, but there exists a whole neighborhood where it bifurcates into two distinct branches.

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arXiv2024

On solutions to a class of degenerate equations with the Grushin operator

The Grushin Laplacian $- Δ_α$ is a degenerate elliptic operator in $\mathbb{R}^{h+k}$ that degenerates on $\{0\} \times \mathbb{R}^k$. We consider weak solutions of $- Δ_αu= Vu$ in an open bounded connected domain $Ω$ with $V \in W^{1,σ}(Ω)$ and $σ> Q/2$, where $Q = h + (1+α)k$ is the so-called homogeneous dimension of $\mathbb{R}^{h+k}$. By means of an Almgren-type monotonicity formula we identify the exact asymptotic blow-up profile of solutions on degenerate points of $Ω$. As an application we derive strong unique continuation properties for solutions.

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arXiv2023

Asymptotic behavior of generalized capacities with applications to eigenvalue perturbations: the higher dimensional case

We provide a full series expansion of a generalization of the so-called $u$-capacity related to the Dirichlet-Laplacian in dimension three and higher, extending previous results of the authors, and of the authors together with Virginie Bonnaillie-Noël, dealing with the planar case. We apply the result in order to study the asymptotic behavior of perturbed eigenvalues when Dirichlet conditions are imposed on a small regular subset of the domain of the eigenvalue problem.

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arXiv2023

Sharp behavior of Dirichlet--Laplacian eigenvalues for a class of singularly perturbed problems

We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of the perturbed eigenvalues. We detect the proper quantity which sharply measures the perturbation's magnitude. It is a sort of torsional rigidity of the tube's section relative to the domain. This allows us to sharply describe the asymptotic behavior of the perturbed spectrum, even when eigenvalues converge to a multiple one. The final asymptotics of eigenbranches depend on the local behavior near the junction of eigenfunctions chosen in a special way. The present techniques also apply when the perturbation of the Dirichlet eigenvalue problem consists in prescribing homogeneous Neumann boundary conditions on a small portion of the boundary of the domain.

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arXiv2022

Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains

Taking advantage from the so-called "Lemma on small eigenvalues" by Colin de Verdière, we study ramification for multiple eigenvalues of the Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior of multiple eigenvalues turns out to depend on the asymptotic expansion of suitable associated eigenfunctions. We treat the case of planar domains in details, thanks to the asymptotic expansion of a generalization of the so-called u-capacity which we compute in dimension 2. In this case multiple eigenvalues are proved to split essentially by different rates of convergence of the perturbed eigenvalues or by different coefficients in front of their expansion if the rate of two eigenbranches turns out to be the same.

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arXiv2019

Asymptotic behavior of $u$-capacities and singular perturbations for the Dirichlet-Laplacian

In this paper 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 planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets $Ω$ and $ω$ of $\mathbb{R}^2$, containing the origin. First, if $\varepsilon$ is positive and small enough and if $u$ is a function defined on $Ω$, we compute an asymptotic expansion of the $u$-capacity $\mathrm{Cap}_Ω(\varepsilon ω, u)$ as $\varepsilon \to 0$. As a byproduct, we compute an asymptotic expansion for the $N$-th eigenvalues of the Dirichlet-Laplacian in the perforated set $Ω\setminus (\varepsilon \overlineω)$ for $\varepsilon$ 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.

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arXiv2019

Eigenvalue variation under moving mixed Dirichlet-Neumann boundary conditions and applications

We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet-Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion's leading term. This allows inferring some remarkable consequences for Aharonov-Bohm eigenvalues when the singular part of the operator has two coalescing poles.

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arXiv2019

On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets

We consider the eigenvalue problem for the restricted fractional Laplacian in a bounded domain with homogeneous Dirichlet boundary conditions. We introduce the notion of fractional capacity for compact subsets, with the property that the eigenvalues are not affected by the removal of zero fractional capacity sets. Given a simple eigenvalue, we remove from the domain a family of compact sets which are concentrating to a set of zero fractional capacity and we detect the asymptotic expansion of the eigenvalue variation; this expansion depends on the eigenfunction associated to the limit eigenvalue. Finally, we study the case in which the family of compact sets is concentrating to a point.

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arXiv2018

On multiple eigenvalues for Aharonov-Bohm operators in planar domains

We study multiple eigenvalues of a magnetic Aharonov-Bohm operator with Dirichlet boundary conditions in a planar domain. In particular, we study the structure of the set of the couples position of the pole-circulation which keep fixed the multiplicity of a double eigenvalue of the operator with the pole at the origin and half-integer circulation. We provide sufficient conditions for which this set is made of an isolated point. The result confirms and validates a lot of numerical simulations available in preexisting literature.

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arXiv2017

Estimates for eigenvalues of Aharonov-Bohm operators with varying poles and non-half-interger circulation

We study the behavior of eigenvalues of a magnetic Aharonov-Bohm operator with non-half-integer circulation and Dirichlet boundary conditions in a planar domain. As the pole is moving in the interior of the domain, we estimate the rate of the eigenvalue variation in terms of the vanishing order of the limit eigenfunction at the limit pole. We also provide an accurate blow-up analysis for scaled eigenfunctions and prove a sharp estimate for their rate of convergence.

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arXiv2016

Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators

We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues of Aharonov-Bohm operators with two colliding poles moving on an axis of symmetry of the domain.

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arXiv2016

Sharp boundary behavior of eigenvalues for Aharonov-Bohm operators with varying poles

In this paper, we investigate the behavior of the eigenvalues of a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a bounded planar domain. We establish a sharp relation between the rate of convergence of the eigenvalues as the singular pole is approaching a boundary point and the number of nodal lines of the eigenfunction of the limiting problem, i.e. of the Dirichlet Laplacian, ending at that point. The proof relies on the construction of a limit profile depending on the direction along which the pole is moving, and on an Almgren-type monotonicity argument for magnetic operators.

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arXiv2015

Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles

We investigate the behavior of eigenvalues for a magnetic Aharonov-Bohm operator with half-integer circulation and Dirichlet boundary conditions in a planar domain. We provide sharp asymptotics for eigenvalues as the pole is moving in the interior of the domain, approaching a zero of an eigenfunction of the limiting problem along a nodal line. As a consequence, we verify theoretically some conjectures arising from numerical evidences in preexisting literature. The proof relies on an Almgren-type monotonicity argument for magnetic operators together with a sharp blow-up analysis.

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