arXiv subjects

Lorenzo Brasco

Publications and source records attributed to Lorenzo Brasco.

At least 19 recordsLinked to original sources

arXiv2026

Regularity for eigenvalue-type equations in sets with thick complement

We prove global higher integrability of the gradient for solutions to nonlinear PDEs with homogeneous Dirichlet condition. We work with general open sets (i.e. not necessarily bounded), under a minimal thickness assumption of the complement. The main focus is on improving the global summability of the gradient, by preserving the ``zero trace" Sobolev class. We pay due attention to providing precise a priori estimates. As an application, we get universal global H\"older estimates for solutions of the super-homogeneous Lane-Emden equation for the $N-$Laplacian, in contractible open sets with finite inradius.

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arXiv2026

Essential spectrum for the $p-$Laplacian

We introduce a variational notion of essential spectrum for the Dirichlet $p-$Laplacian. We then extend the classical Persson Theorem to this nonlinear setting. This result provides a geometric characterization of the bottom of the essential spectrum, in terms of the sharp $L^p$ Poincaré constant ``at infinity''. We also show that in the case $p=2$ our construction of the essential spectrum is perfectly consistent with the classical theory. Finally, as an example, we compute the full spectrum of the Dirichlet $p-$Laplacian on a rectilinear strip: it is purely essential, with no embedded eigenvalues. The arguments of the proofs are elementary and new already for the linear case $p=2$.

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arXiv2026

Low eigenvalues of the $p-$Laplacian in general open sets

We consider the minmax Ljusternik-Schnirelmann levels of the constrained $p-$Dirichlet integral, on a general open set of the Euclidean space. We show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincaré constant ``at infinity'', it actually defines an eigenvalue of the Dirichlet $p-$Laplacian. We also prove an exponential decay at infinity for the relevant eigenfunctions: this can be seen as a Šnol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.

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arXiv2025

Extremals for sharp Poincaré-Sobolev inequalities: periodically perforated sets and beyond

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincaré-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.

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arXiv2025

Extremals for Poincaré-Sobolev sharp constants in Steiner symmetric sets

We prove existence of minimizers for the sharp Poincaré-Sobolev constant in general Steiner symmetric sets, in the subcritical and superhomogeneous regime. The sets considered are not necessarily bounded, thus the relevant embeddings may suffer from a lack of compactness. We prove existence by means of an elementary compactness method. We also prove an exponential decay at infinity for minimizers, showing that in the case of Steiner symmetric sets the relevant estimates only depend on the underlying geometry. Finally, we illustrate the optimality of the existence result, by means of some examples.

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arXiv2025

A Schrödinger operator with confining potential having quadratic growth

We study the spectral properties of a Schrödinger operator, in presence of a confining potential given by the distance squared from a fixed compact potential well. We prove continuity estimates on both the eigenvalues and the eigenstates, lower bounds on the ground state energy, regularity and integrability properties of eigenstates. We also get explicit decay estimates at infinity, by means of elementary nonlinear methods.

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arXiv2025

Variations on the capacitary inradius

We discuss some properties of the capacitary inradius for an open set. This is an extension of the classical concept of inradius (i.e. the radius of a largest inscribed ball), which takes into account capacitary effects. Its introduction dates back to the pioneering works of Vladimir Maz'ya. We present some variants of this object and their mutual relations, as well as their connections with Poincaré inequalities. We also show that, under a mild regularity assumption on the boundary of the sets, the capacitary inradius is equivalent to the classical inradius. This comes with an explicit estimate and it permits to get a Buser-type inequality for a large class of open sets, whose boundaries may have power-like cusps of arbitrary order. Finally, we present a couple of open problems.

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arXiv2024

Capacitary inradius and Poincaré-Sobolev inequalities

We prove a two-sided estimate on the sharp $L^p$ Poincaré constant of a general open set, in terms of a capacitary variant of its inradius. This extends a result by Maz'ya and Shubin, originally devised for the case $p=2$, in the subconformal regime. We cover the whole range of $p$, by allowing in particular the extremal cases $p=1$ (Cheeger's constant) and $p=N$ (conformal case), as well. We also discuss the more general case of the sharp Poincaré-Sobolev embedding constants and get an analogous result. Finally, we present a brief discussion on the superconformal case, as well as some examples and counter-examples.

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arXiv2024

Beyond Cheeger's constant

The Cheeger constant of an open set of the Euclidean space is defined by minimizing the ratio "perimeter over volume", among all its smooth compactly contained subsets. We consider a natural variant of this problem, where the volume of admissible sets is raised to any positive power. We show that for {\it sublinear} powers, all these generalized Cheeger constants are equivalent to the standard one, by means of a universal two-sided estimate. We also show that this equivalence breaks down for {\it superlinear} powers. In this case, some weird phenomena appear. We finally consider the case of convex planar sets and prove an existence result, under optimal assumptions.

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arXiv2024

The role of topology and capacity in some bounds for principal frequencies

We prove a lower bound on the sharp Poincaré-Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the restriction that points are not removable sets. In the first case, we get an estimate which optimally depends on the topology of the sets, thus generalizing a result by Croke, Osserman and Taylor, originally devised for the first eigenvalue of the Dirichlet-Laplacian. We also consider some limit situations, like the sharp Moser-Trudinger constant and the Cheeger constant. As a a byproduct of our discussion, we also obtain a Buser--type inequality for open subsets of the plane, with given topology. An interesting problem on the sharp constant for this inequality is presented.

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arXiv2023

On Morrey's inequality in Sobolev-Slobodecki\uı spaces

We study the sharp constant in the Morrey inequality for fractional Sobolev-Slobodecki\uı spaces on the whole $\mathbb{R}^N$. By generalizing a recent work by Hynd and Seuffert, we prove existence of extremals, together with some regularity estimates. We also analyze the sharp asymptotic behaviour of this constant as we reach the borderline case $s\,p=N$, where the inequality fails. This can be done by means of a new elementary proof of the Morrey inequality, which combines: a local fractional Poincaré inequality for punctured balls, the definition of capacity of a point and Hardy's inequality for the punctured space. Finally, we compute the limit of the sharp Morrey constant for $s\nearrow 1$, as well as its limit for $p\nearrow \infty$. We obtain convergence of extremals, as well.

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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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arXiv2023

Sobolev embeddings and distance functions

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the superconformal case (i.e. when $p$ is larger than the dimension) these two facts are equivalent, while in the subconformal and conformal cases (i.e. when $p$ is less than or equal to the dimension) we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behaviour of the positive solution of the Lane-Emden equation for the $p-$Laplacian with sub-homogeneous right-hand side, as the exponent $p$ diverges to $\infty$. The case of first eigenfunctions of the $p-$Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.

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arXiv2023

An optimal lower bound in fractional spectral geometry for planar sets with topological constraints

We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman and Taylor, in the limit as $s$ goes to $1$. The limit as $s$ goes to $1/2$ is carefully analyzed, as well.

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arXiv2022

Singular orthotropic functionals with nonstandard growth conditions

We pursue the study of a model convex functional with orthotropic structure and nonstandard growth conditions, this time focusing on the sub-quadratic case. We prove that bounded local minimizers are locally Lipschitz. No restriction on the ratio between the highest and the lowest growth rates are needed. The result holds also in presence of a non-autonomous lower order term, under sharp integrability assumptions. Finally, we prove higher differentiability of bounded local minimizers, as well.

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arXiv2022

Uniqueness of extremals for some sharp Poincaré-Sobolev constants

We study the sharp constant for the embedding of $W^{1,p}_0(Ω)$ into $L^q(Ω)$, in the case $2 p$ and $q$ is sufficiently close to $p$, extremal functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side. The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace--type equations by L. Damascelli and B. Sciunzi.

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arXiv2022

A note on the supersolution method for Hardy's inequality

We prove a characterization of Hardy's inequality in Sobolev-Slobodecki\uı spaces in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation. This extends previous results by Ancona and Kinnunen & Korte for standard Sobolev spaces. The proof is based on variational methods.

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arXiv2022

On the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces

We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then restrict the discussion to open convex sets and compute such a sharp constant, by constructing suitable supersolutions by means of the distance function. Such a method of proof works only for $s\,p\ge 1$ or for $Ω$ being a half-space. We exhibit a simple example suggesting that this method can not work for $s\,p<1$ and $Ω$ different from a half-space. The case $s\,p<1$ for a generic convex set is left as an interesting open problem, except in the Hilbertian setting (i.e. for $p=2$): in this case we can compute the sharp constant in the whole range $0<s<1$. This completes a result which was left open in the literature.

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