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Roberto Ognibene

Publications and source records attributed to Roberto Ognibene.

At least 19 recordsLinked to original sources
arXiv2026

On the stability of eigenvalues of varying bilinear forms in abstract Hilbertian settings and applications

The aim of the present paper is to develop a spectral perturbation theory from a higher perspective. More precisely, we consider a one-parameter family of varying bilinear forms, each of them defined on a (possibly) different Hilbert space. Assuming the stability of the corresponding spectra, our first main result establishes a quantification of the rate of convergence. A key feature is the explicit variational characterization of the first term in the asymptotic expansion of the perturbed eigenvalues, which only depends on the ``data'', i.e. the limit eigenspace and the magnitude of the perturbation, through the resolution of a minimization problem. Remarkably, we make no assumptions on the perturbed eigenelements (besides, naturally, the spectral stability). Moreover, we cover both the cases of simple and multiple limit eigenvalues in full generality. In the second part, we explore some concrete applications of our abstract results. First, we consider eigenvalue problems for the Laplace-Beltrami operator with varying measure weights (also motivated by optimization in spectral geometry); secondly, we investigate the Neumann approximation of the Steklov eigenvalues of the Laplacian; finally, we focus on how the spectrum of the Laplace-Beltrami operator on a Riemannian manifold changes when a second small manifold is glued on a small portion of it.

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arXiv2026

On fractional critical problems with multi-polar Hardy potentials

We investigate the existence of positive solutions to fractional equations presenting a double criticality: a multi-polar Hardy-type potential and a Sobolev critical nonlinearity. The nonlocal nature of the operator and the absence of explicit ground states for the single-pole equation stand as major difficulties. We overcome these obstacles by passing to an extended formulation of the problem and by establishing sharp asymptotic estimates for the solutions in the case of a single pole. Then, through a concentration-compactness argument, we show that the existence of minimizers is dictated by the magnitude of the masses and the mutual distances between the corresponding poles.

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arXiv2026

On the splitting of Neumann eigenvalues in perforated domains

We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. Our main result establishes that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension $N-1$ and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter. Such an asymptotic formula is of independent interest and generalizes previous results; notably, in dimension $N\geq 3$, it is valid for holes of arbitrary shape.

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arXiv2025

A survey on the optimal partition problem

This survey synthesizes the current state of the art on the regularity theory for solutions to the optimal partition problem. Namely, we consider non-negative, vector-valued Sobolev functions whose components have mutually disjoint support, and which are either local minimizers of the Dirichlet energy or, more generally, critical points satisfying a system of variational inequalities. This is particularly meaningful as the problem has emerged on several occasions and in diverse contexts: our aim is then to provide a coherent point of view and an up-to-date account of the progress concerning regularity of the solutions and their free boundaries, both in the interior and up to a fixed boundary.

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arXiv2025

Asymptotics of nonlinear Robin energies

This paper investigates the asymptotic behavior of a class of nonlinear variational problems with Robin-type boundary conditions on a bounded Lipschitz domain. The energy functional contains a bulk term (the $p$-norm of the gradient), a boundary term (the $q$-norm of the trace) scaled by a parameter $\alpha>0$, and a linear source term. By variational methods, we derive first-order expansions of the minimum as $\alpha\to 0^+$ (Neumann limit) and as $\alpha\to+\infty$ (Dirichlet limit). In the Dirichlet limit, the energy converges to the one of Dirichlet problem with a power-type quantified rate (depending only on $q$), while the Neumann limit exhibits a dichotomy: under a compatibility condition, the energy linearly approaches the one of Neumann problem, otherwise, it diverges as a power of $\alpha$ depending only on $q$.

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arXiv2024

Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems

We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency $3/2$, the free interface is composed of three $C^{1,\alpha}$-smooth $(d-1)$-dimensional manifolds (composed of points of frequency $1$) with common $C^{1,\alpha}$-regular boundary (made of points of frequency $3/2$) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet eigenvalues.

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arXiv2024

On asymptotics of Robin eigenvalues in the Dirichlet limit

We investigate the asymptotic behavior of the eigenvalues of the Laplacian with homogeneous Robin boundary conditions, when the (positive) Robin parameter is diverging. In this framework, since the convergence of the Robin eigenvalues to the Dirichlet ones is known, we address the question of quantifying the rate of such convergence. More precisely, in this work we identify the proper geometric quantity representing (asymptotically) the first term in the expansion of the eigenvalue variation: it is a novel notion of torsional rigidity. Then, by performing a suitable asymptotic analysis of both such quantity and its minimizer, we prove the first-order expansion of any Robin eigenvalue, in the Dirichlet limit. Moreover, the convergence rate of the corresponding eigenfunctions is obtained as well. We remark that all our spectral estimates are explicit and sharp, and cover both the cases of convergence to simple and multiple Dirichlet eigenvalues.

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arXiv2024

Boundary regularity of the free interface in spectral optimal partition problems

We consider the problem of optimal partition of a domain with respect to the sum of the principal eigenvalues and we prove for the first time regularity results for the free interface up to fixed boundary. All our results are quantitative and, in particular, we obtain fine estimates on the continuity of the solutions and the oscillation of the free interface (in terms of the modulus of continuity of the normal vector of the fixed boundary), even in the case of domains with low (Dini-type) regularity. Our analysis is based on an Almgren-type monotonicity formula at boundary points and an epiperimetric inequality at points of low frequency, which, together, yield an explicit rate of convergence for blow-up sequences and the boundary strong unique continuation property. Exploiting our quantitative blow-up analysis, we manage to prove clean-up results near one-phase and two-phase points. We define the notion of free interface inside the fixed boundary, and we prove that the subset of points of minimal frequency is regular and that the interior free interface is approaching the boundary orthogonally in a smooth way.

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arXiv2023

Quantitative spectral stability for Aharonov-Bohm operators with many coalescing poles

The behavior of simple eigenvalues of Aharonov-Bohm operators with many coalescing poles is discussed. In the case of half-integer circulation, a gauge transformation makes the problem equivalent to an eigenvalue problem for the Laplacian in a domain with straight cracks, laying along the moving directions of poles. For this problem, we obtain an asymptotic expansion for eigenvalues, in which the dominant term is related to the minimum of an energy functional associated with the configuration of poles and defined on a space of functions suitably jumping through the cracks. Concerning configurations with an odd number of poles, an accurate blow-up analysis identifies the exact asymptotic behaviour of eigenvalues and the sign of the variation in some cases. An application to the special case of two poles is also discussed.

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arXiv2023

Quantitative spectral stability for the Neumann Laplacian in domains with small holes

The aim of the present paper is to investigate the behavior of the spectrum of the Neumann Laplacian in domains with little holes excised from the interior. More precisely, we consider the eigenvalues of the Laplacian with homogeneous Neumann boundary conditions on a bounded, Lipschitz domain. Then, we singularly perturb the domain by removing Lipschitz sets which are "small" in a suitable sense and satisfy a uniform extension property. In this context, we provide an asymptotic expansion for all the eigenvalues of the perturbed problem which are converging to simple eigenvalues of the limit one. The eigenvalue variation turns out to depend on a geometric quantity resembling the notion of (boundary) torsional rigidity: understanding this fact is one of the main contributions of the present paper. In the particular case of a hole shrinking to a point, through a fine blow-up analysis, we identify the exact vanishing order of such a quantity and we establish some connections between the location of the hole and the sign of the eigenvalue variation.

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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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arXiv2023

On the spectrum of sets made of cores and tubes

We analyze the spectral properties of a particular class of unbounded open sets. These are made of a central bounded ``core'', with finitely many unbounded tubes attached to it. We adopt an elementary and purely variational point of view, studying the compactness (or the defect of compactness) of level sets of the relevant constrained Dirichlet integral. As a byproduct of our argument, we also get exponential decay at infinity of variational eigenfunctions. Our analysis includes as a particular case a planar set (sometimes called ``bookcover''), already encountered in the literature on curved quantum waveguides. J. Hersch suggested that this set could provide the sharp constant in the {\it Makai-Hayman inequality} for the bottom of the spectrum of the Dirichlet-Laplacian of planar simply connected sets. We disprove this fact, by means of a singular perturbation technique.

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arXiv2022

Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Neumann region

We deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region.

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arXiv2022

On a weighted two-phase boundary obstacle problem

In this work we consider an inhomogeneous two-phase obstacle-type problem driven by the fractional Laplacian. In particular, making use of the Caffarelli-Silvestre extension, Almgren and Monneau type monotonicity formulas and blow-up analysis, we provide a classification of the possible vanishing orders, which implies the strong unique continuation property. Moreover, we prove a stratification result for the nodal set, together with estimates on its Hausdorff dimensions, for both the regular and the singular part. The main tools come from geometric measure theory and amount to Whitney's Extension Theorem and Federer's Reduction Principle.

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arXiv2022

Asymptotic behavior of constrained local minimizers in finite elasticity

We provide an approximation result for the pure traction problem of linearized elasticity in terms of local minimizers of finite elasticity, under the constraint of vanishing average curl for admissible deformation maps. When suitable rotations are included in the constraint, the limit is shown to be the linear elastic equilibrium associated to rotated loads.

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arXiv2021

A quantitative stability inequality for fractional capacities

The aim of this work is to show a non-sharp quantitative stability version of the fractional isocapacitary inequality. In particular, we provide a lower bound for the isocapacitary deficit in terms of the Fraenkel asymmetry. In addition, we provide the asymptotic behaviour of the $s$-fractional capacity when $s$ goes to $1$ and the stability of our estimate with respect to the parameter $s$.

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arXiv2020

Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region

In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit Neumann problem. The first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. Then we focus on the case of Dirichlet boundary conditions imposed on a subset which is scaling to a point; by a blow-up analysis for the capacitary potentials, we detect the vanishing order of the Sobolev capacity of such shrinking Dirichlet boundary portion.

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