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Luigi Provenzano

Publications and source records attributed to Luigi Provenzano.

At least 37 records · Page 2Linked to original sources
arXiv2024

Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator

Given a compact manifold $\mathcal M$ with boundary of dimension $n\geq 3$ and any integers $K$ and $N$, we show that there exists a metric on $\mathcal M$ for which the first $K$ nonconstant eigenfunctions of the Dirichlet-to-Neumann map on $\partial\mathcal M$ have at least $N$ nodal components. This provides a negative answer to the question of whether the number of nodal domains of Dirichlet-to-Neumann eigenfunctions satisfies a Courant-type bound, which has been featured in recent surveys by Girouard and Polterovich [21, Open problem 9] and by Colbois, Girouard, Gordon and Sher [9, Open question 10.14].

math.SP↗
arXiv2023

Geometry of the magnetic Steklov problem on Riemannian annuli

We study the geometry of the first two eigenvalues of a magnetic Steklov problem on an annulus $Σ$ (a compact Riemannian surface with genus zero and two boundary components), the magnetic potential being the harmonic one-form having flux $ν\in\mathbb R$ around any of the two boundary components. The resulting spectrum can be seen as a perturbation of the classical, non-magnetic Steklov spectrum, obtained when $ν=0$ and studied e.g., by Fraser and Schoen. We obtain sharp upper bounds for the first and the second normalized eigenvalues and we discuss the geometry of the maximisers. Concerning the first eigenvalue, we isolate a noteworthy class of maximisers which we call $α$-surfaces: they are free-boundary surfaces which are stationary for a weighted area functional (depending on the flux) and have proportional principal curvatures at each point; in particular, they belong to the class of linear Weingarten surfaces. We then study the second normalized eigenvalue for a fixed flux $ν$ and prove the existence of a maximiser for rotationally invariant metrics. Moreover, the corresponding eigenfunctions define a free-boundary immersion in the unit ball of $\mathbb R^3$. Finally, we prove that the second normalized eigenvalue associated to a flux $ν$ has an absolute maximum when $ν=0$, the corresponding maximiser being the critical catenoid.

math.SP↗
arXiv2023

On the critical points of semi-stable solutions on convex domains of Riemannian surfaces

In this paper we consider semilinear equations $-Δu=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has exactly one non-degenerate critical point (a maximum). The proof consists in relating the critical points of the solution with the critical points of a suitable auxiliary function, jointly with a topological degree argument.

math.DG↗
arXiv2023

Geometric bounds for the magnetic Neumann eigenvalues in the plane

We consider the eigenvalues of the magnetic Laplacian on a bounded domain $Ω$ of $\mathbb R^2$ with uniform magnetic field $β>0$ and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy $λ_1$ and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields $β=β(x)$ on a simply connected domain in a Riemannian surface. In particular: we prove the upper bound $λ_1<β$ for a general plane domain, and the upper bound $λ_1<\sup_{x\inΩ}|β(x)|$ for a variable magnetic field when $Ω$ is simply connected. For smooth domains, we prove a lower bound of $λ_1$ depending only on the intensity of the magnetic field $β$ and the rolling radius of the domain. The estimates on the Riesz mean imply an upper bound for the averages of the first $k$ eigenvalues which is sharp when $k\to\infty$ and consists of the semiclassical limit $\dfrac{2πk}{|Ω|}$ plus an oscillating term. We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which $λ_1$ is always small.

math.SP↗
arXiv2023

Semiclassical estimates for eigenvalue means of Laplacians on spheres

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for Pólya-type inequalities. Finally, we provide sum rules for Laplacian eigenvalues on spheres and compact two-point homogeneous spaces.

math.SP↗
arXiv2022

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

Let $Ω$ be a bounded domain in $\mathbb R^2$ with smooth boundary $\partialΩ$, and let $ω_h$ be the set of points in $Ω$ whose distance from the boundary is smaller than $h$. We prove that the eigenvalues of the biharmonic operator on $ω_h$ with Neumann boundary conditions converge to the eigenvalues of a limiting problem in the form of system of differential equations on $\partialΩ$.

math.SP↗
arXiv2022

The first Grushin eigenvalue on cartesian product domains

In this paper we consider the first eigenvalue $λ_1(Ω)$ of the Grushin operator $Δ_G:=Δ_{x_1}+|x_1|^{2s}Δ_{x_2}$ with Dirichlet boundary conditions on a bounded domain $Ω$ of $\mathbb{R}^d= \mathbb{R}^{d_1+d_2}$. We prove that $λ_1(Ω)$ admits a unique minimizer in the class of domains with prescribed finite volume which are the cartesian product of a set in $\mathbb{R}^{d_1}$ and a set in $\mathbb{R}^{d_2}$, and that the minimizer is the product of two balls $Ω^*_1 \subseteq \mathbb{R}^{d_1}$ and $Ω_2^* \subseteq \mathbb{R}^{d_2}$. Moreover, we provide a lower bound for $|Ω^*_1|$ and for $λ_1(Ω_1^*\timesΩ_2^*)$. Finally, we consider the limiting problem as $s$ tends to $0$ and to $+\infty$.

math.AP↗
arXiv2022

Isoperimetric inequalities for the magnetic Neumann and Steklov problems with Aharonov-Bohm magnetic potential

We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We establish isoperimetric inequalitites for the lowest eigenvalue in the spirit of the classical inequalities of Szegö-Weinberger, Brock and Weinstock, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which include the spherical and the hyperbolic case.

math.SP↗
arXiv2021

Isoparametric foliations and the Pompeiu problem

A bounded domain $Ω$ in a Riemannian manifold $M$ is said to have the Pompeiu property if the only continuous function which integrates to zero on $Ω$ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $M$ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $M$ under which the level domains of an isoparametric function fail the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.

math.DG↗
arXiv2021

Semiclassical bounds for spectra of biharmonic operators

We provide complementary semiclassical bounds for the Riesz means $R_1(z)$ of the eigenvalues of various biharmonic operators, with a second term in the expected power of $z$. The method we discuss makes use of the averaged variational principle (AVP), and yields two-sided bounds for individual eigenvalues, which are semiclassically sharp. The AVP also yields comparisons with Riesz means of different operators, in particular Laplacians.

math.SP↗
arXiv2021

On the spectral asymptotics for the buckling problem

We provide a direct proof of Weyl's law for the buckling eigenvalues of the biharmonic operator on a wide class of domains of $\mathbb R^d$ including bounded Lipschitz domains. The proof relies on asymptotically sharp lower and upper bounds that we develop for the Riesz mean $R_2(z)$. Lower bounds are obtained by making use of the so-called "averaged variational principle". Upper bounds are obtained in the spirit of Berezin-Li-Yau. Moreover, we state a conjecture for the second term in Weyl's law and prove its correctness in two special cases: balls in $\mathbb R^d$ and bounded intervals in $\mathbb R$.

math.SP↗
arXiv2021

Upper bounds for the Steklov eigenvalues of the $p$-Laplacian

In this note we present upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian on domains of $\mathbb R^n$, $n\geq 2$. We show that for $1 n$ upper bounds depend on a geometric constant $D(Ω)$, the $(n-1)$-distortion of $Ω$ which quantifies the concentration of the boundary measure. We prove that the presence of this constant is necessary in the upper estimates for $p>n$ and that the corresponding inequality is sharp, providing examples of domains with boundary measure uniformly bounded away from zero and infinity and arbitrarily large variational eigenvalues.

math.SP↗
arXiv2020

Conformal upper bounds for the eigenvalues of the $p$-Laplacian

In this note we present upper bounds for the variational eigenvalues of the $p$-Laplacian on smooth domains of complete $n$-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given manifold $(M,g)$ for $1 1$ when we fix a metric $g$. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.

math.SP↗
arXiv2019

On Trace Theorems for Sobolev Spaces

We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov's Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space $H^1(Ω)$ on a Lipschitz domain $Ω$, and which we have further developed for the trace spaces of $H^k(Ω)$, $k\geq 2$, by using Fourier expansions associated with the eigenfunctions of new multi-parameter polyharmonic Steklov problems.

math.FA↗
arXiv2019

On the explicit representation of the trace space $H^{\frac{3}{2}}$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

We consider the problem of describing the traces of functions in $H^2(Ω)$ on the boundary of a Lipschitz domain $Ω$ of $\mathbb R^N$, $N\geq 2$. We provide a definition of those spaces, in particular of $H^{\frac{3}{2}}(\partialΩ)$, by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $H^1(Ω)$, based on the classical second order Steklov problem.

math.SP↗
arXiv2019

Complementary asymptotically sharp estimates for eigenvalue means of Laplacians

We present asymptotically sharp inequalities, containing a second term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kröger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean $R_1(z)$ of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.

math.SP↗