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Luigi Provenzano

Publications and source records attributed to Luigi Provenzano.

At least 19 recordsLinked to original sources
arXiv2026

Remarks on Weyl-type bounds for Steklov eigenvalues

We prove that, for $n\geq 3$, there is no constant $C_n>0$ depending only on $n$ such that the Steklov eigenvalues $σ_k(Ω)$ satisfy $|\partialΩ|^{\frac 1{n-1}}σ_k(Ω)\leq C_nk^{\frac1{n-1}}$ for every smooth bounded domain $Ω\subset\mathbb R^n$ and every $k\geq1 $, providing a negative answer to an open problem posed by Girouard and Polterovich \cite{GiPo}. On the other hand, we prove that there exists a constant $C_n>0$ depending only on $n$ such that $|Ω|^{\frac 1n}σ_k(Ω)\leq C_nk^{\frac1{n-1}}$ for every smooth bounded domain $Ω\subset\mathbb R^n$ and every $k\geq 1$. This estimate, combined with the isoperimetric bound of Colbois, El Soufi and Girouard \cite{colboisgirouard_steklov}, implies the bound $|\partialΩ|^{\frac 1{n-1}}σ_k(Ω)\leq C_nk^{\frac1{n-1}+\frac{n-2}{n(n-1)^2}}$, where the exponent of $k$ turns out to be sharp.

math.SP↗
arXiv2026

Regularity results for elliptic equations on cones

We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $λ_1(D)\ge N-1$, where $λ_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-Δ_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.

math.AP↗
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

Lane-Emden Problems on Convex Domains of $\mathbb S^2$

We study positive solutions of the Dirichlet problem $-Δu = u^p$ in a uniformly convex domain $Ω\subset \mathbb S^2$, $u= 0$ on $\partialΩ.$ For $p=1$, we assume that the right-hand side is replaced by $λ_1 u$, where $λ_1$ is the first eigenvalue of $-Δ$ on $Ω$ with zero Dirichlet boundary condition. We prove that for $0 \leq p < 1$ the unique positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly concave in $Ω$, while for $1 < p \leq 3$ every positive solution $u$ is such that $u^{\frac{1-p}{2}}$ is strictly convex in $Ω.$ For $p=0,$ our result gives the strict $1/2-$concavity of the torsion function in $Ω.$ For $p=1,$ a result due to Lee and Wang gives the strict log-concavity of the first eigenfunction in $Ω.$ As a consequence, for each $0 \leq p \leq 3,$ any positive solution has strictly convex superlevel sets and a unique nondegenerate maximum.

math.AP↗
arXiv2026

Isoperimetric inequalities and sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces

We consider the first eigenvalue of the magnetic Laplacian with zero magnetic field on simply connected compact surfaces and we establish isoperimetric inequalities and upper bounds in terms of a bound on the gaussian curvature. As a corollary, we prove that among all simply connected spherical domains of fixed area, the first eigenvalue is maximal for a geodesic disk with the pole of the magnetic potential at its center; also, for the sphere punctured at two points, the first eigenvalue is maximal when the punctures are antipodal.

math.SP↗
arXiv2026

A note on the Maxwell's eigenvalues on thin sets

We analyse the Maxwell's spectrum on thin tubular neighborhoods of embedded surfaces of $\mathbb R^3$. We show that the Maxwell eigenvalues converge to the Laplacian eigenvalues of the surface as the thin parameter tends to zero. To achieve this, we reformulate the problem in terms of the spectrum of the Hodge Laplacian with relative conditions acting on co-closed differential $1$-forms. The result leads to new examples of domains where the Faber-Krahn inequality for Maxwell's eigenvalues fails, examples of domains with any number of arbitrarily small eigenvalues, and underlines the failure of spectral stability under singular perturbations changing the topology of the domain. Additionally, we explicitly produce the Maxwell's eigenfunctions on product domains with the product metric, extending previous constructions valid in the Euclidean case.

math.SP↗
arXiv2026

Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation

We prove the non-degeneracy of solutions to a fractional and singular Liouville equation defined on the whole real line in presence of a singular term. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a bounded domain, which is defined either by an intersection or a union of two disks. We conclude by proving the simplicity of the corresponding eigenvalue.

math.AP↗
arXiv2025

Bifurcation from bubbles in nonconvex cones

We investigate the Neumann problem for the critical semilinear elliptic equation in cones. The standard bubble provides a family of radial solutions, which are known to be the only positive solutions in convex cones. For nonconvex cones, symmetry breaking may occur and the symmetry breaking is related to the first nonzero Neumann eigenvalue of the Laplace Beltrami operator on the domain $D\subset§^{N-1}$, that spans the cone. We construct a one-parameter family of domains on the sphere whose first eigenvalue crosses the threshold at which the bubble loses stability. Under the assumption that this eigenvalue is simple, we prove, via the Crandall Rabinowitz bifurcation theorem, the existence of a branch of nonradial solutions bifurcating from the standard bubble. Moreover we show that the bifurcation is global.

math.AP↗
arXiv2025

The role of the curvature of a surface in the shape of the solutions to elliptic equations

We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-Δu=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal.

math.AP↗
arXiv2025

A note on the first Steklov eigenvalue on planar domains

We consider the first positive Steklov eigenvalue on planar domains. First, we provide an example of a planar domain for which a first eigenfunction has a closed nodal line. Second, we establish a lower bound for the first positive eigenvalue on certain symmetric domains and show that this eigenvalue is simple for all ellipses. These results complement two statements contained in a work by Kuttler and Sigillito (Proc. Amer. Math. Soc. 20, 1969).

math.AP↗
arXiv2025

Magnetic ground states and the conformal class of a surface

On a closed, orientable Riemannian surface $Σ_g$ of arbitrary genus $g\geq 1$ and Riemannian metric $h$ we study the magnetic Laplacian with magnetic potential given by a harmonic $1$-form $A$. Its lowest eigenvalue (magnetic ground state energy) is positive, unless $A$ represents an integral cohomology class. We isolate a countable set of ground state energies which we call $\textit{ground state spectrum}$ of the metric $h$. The main result of the paper is to show that the ground state spectrum determines the volume and the conformal class of the metric $h$. In particular, hyperbolic metrics are distinguished by their ground state spectrum. We also compute the magnetic spectrum of flat tori and introduce some magnetic spectral invariants of $(Σ_g,h)$ which are conformal by definition and involve the geometry of what we call the Jacobian torus of $(Σ_g,h)$ (in Algebraic Geometry, the Jacobian variety of a Riemann surface).

math.DG↗
arXiv2025

Semiclassical eigenvalue bounds for compact homogeneous irreducible Riemannian manifolds

We exploit an identity for the gradients of Laplacian eigenfunctions on compact homogeneous Riemannian manifolds with irreducible linear isotropy group to obtain asymptotically sharp universal eigenvalue inequalities and sharp Weyl bounds on Riesz means. The approach is non variational and is based on identities for spectral quantities in the form of sum rules.

math.SP↗
arXiv2024

A reverse Faber-Krahn inequality for the magnetic Laplacian

We consider the first eigenvalue of the magnetic Laplacian in a bounded and simply connected planar domain, with uniform magnetic field and Neumann boundary conditions. We investigate the reverse Faber-Krahn inequality conjectured by S. Fournais and B. Helffer, stating that this eigenvalue is maximized by the disk for a given area. Using the method of level lines, we prove the conjecture for small enough values of the magnetic field (those for which the corresponding eigenfunction in the disk is radial).

math.SP↗
arXiv2024

On the critical points of Steklov eigenfunctions

We consider the critical points of Steklov eigenfunctions on a compact, smooth $n$-dimensional Riemannian manifold $M$ with boundary $\partial M$. For generic metrics on $M$ we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to $\partial M$, and the Euler characteristic of $M$. In dimension $2$ this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of $M$ and of the number of sign changes of $u$ on $\partial M$. In the case of the second Steklov eigenfunction on a genus $0$ surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on $M$ Steklov eigenfunctions are Morse functions in $M$.

math.AP↗
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↗