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Alessandro Savo

Publications and source records attributed to Alessandro Savo.

At least 19 recordsLinked to original sources
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.

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arXiv2025

Rigidity of an overdetermined heat equation and minimal helicoids in space-forms

Let $M$ be a Riemannian manifold and $\Omega$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $\Omega$, and is $0$ on its complement. Heat will then flow away from $\Omega$ to its complement, and we are interested in the temperature on the boundary of $\Omega$ at all positive times $t>0$. In particular we ask: are there domains for which the temperature at the boundary is a constant $c$, for all positive times $t$ and for all points of the boundary? If they exist, what can we say about their geometry? This is a typical example of overdetermined heat equation. It is readily seen that if $c$ exists it must be $\frac 12$, and domains with constant boundary temperature will be said to have the $\frac 12$-property. Previous work by \cite{MPS06} and \cite{CSU23} show that, on $\mathbb R^3$, the only such domains (up to congruences) have boundary which is a plane or (a bit surprisingly) the right helicoid. In this paper we first show that, in great generality, the boundary of a $\frac 12$-domain must be minimal; we then extend (with a different proof) the above classification from $\mathbb R^3$ to the other $3$-dimensional space-forms. We prove that, in $\mathbb S^3$, $\frac 12$-domains are bounded by a totally geodesic surface or the Clifford torus, and in the hyperbolic space $\mathbb H^3$ are bounded by a totally geodesic surface or by an (embedded) minimal hyperbolic helicoid. %(there is a one-parameter family of such surfaces) As a by-product, we extend (with a different proof) a result by Nitsche on uniformly dense domains from $\mathbb R^3$ to $3$-dimensional space-forms.

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arXiv2025

Magnetic ground states and the conformal class of a surface

On a closed, orientable Riemannian surface $\Sigma_g$ 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 $(\Sigma_g,h)$ which are conformal by definition and involve the geometry of what we call the Jacobian torus of $(\Sigma_g,h)$ (in Algebraic Geometry, the Jacobian variety of a Riemann surface).

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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).

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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.

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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.

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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.

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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.

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arXiv2021

On the heat content functional and its critical domains

We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that a domain is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established by the author. The outcome is that a domain is critical for the heat content at all times if and only if it admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have constant mean curvature. Then, we consider the sequence of functionals given by the exit-time moments, which generalize the torsional rigidity. We prove that a domain is critical for all exit time moments if and only it is critical for the heat content at all times, and then we get a classification as well. The main purpose of the paper is to understand the variational properties of general isoparametric foliations and their role in PDE's theory; in some respect isoparametric foliations generalize the properties of the foliation of Euclidean space by round spheres.

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arXiv2020

Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential $1$-form (hence, with zero magnetic field) acting on complex functions of a planar domain $Ω$, with magnetic Neumann boundary conditions. It is well-known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $ε$-net. In the last part we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.

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arXiv2020

Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains

We study the Laplacian with zero magnetic field acting on complex functions of a planar domain $Ω$, with magnetic Neumann boundary conditions. If $Ω$ is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of $Ω$. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the the fluxes of the potential one-form around the inner holes and the distance between the boundary components of the domain; more precisely, the ratio between its minimal and maximal width. Then, we give a lower bound for doubly connected domains which is sharp in terms of this ratio, and a general lower bound for domains with an arbitrary number of holes. When the inner holes shrink to points, we obtain as a corollary a lower bound for the first eigenvalue of the so-called Aharonov-Bohm operators with an arbitrary number of poles.

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arXiv2019

Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

We consider the first eigenvalue $λ_1(Ω,σ)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $Ω$ with smooth boundary, $σ\in\bf R$ being the Robin boundary parameter. When $σ>0$ we give a positive, sharp lower bound of $λ_1(Ω,σ)$ in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of $Ω$, a lower bound of the mean curvature of $\partialΩ$ and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean domains are obtained, which improve known estimates. Then, we extend a monotonicity result for $λ_1(Ω,σ)$ obtained in Euclidean space by Giorgi and Smits to a class of manifolds of revolution which include all space forms of constant sectional curvature. As an application, we prove that $λ_1(Ω,σ)$ is uniformly bounded below by $\frac{(n-1)^2}4$ for all bounded domains in the hyperbolic space of dimension $n$, provided that the boundary parameter $σ\geq\frac{n-1}{2}$ (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed

math.AP↗
arXiv2018

Geometric rigidity of constant heat flow

Let $Ω$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $Ω$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is a constant function on the boundary, we say that $Ω$ has the {\it constant flow property}. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that $Ω$ has the constant flow property if and only if it is an {\it isoparametric tube}, that is, it is a solid tube of constant radius around a closed, smooth, minimal submanifold, with the additional property that all equidistants to the boundary (parallel hypersurfaces) are smooth and have constant mean curvature. Hence, the constant flow property can be viewed as an analytic counterpart to the isoparametric property. Finally, we relate the constant flow property with other overdetermined problems, in particular, the well-known Serrin problem on the mean-exit time function, and discuss a counterexample involving minimal free boundary immersions into Euclidean balls.

math.DG↗
arXiv2018

Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square summable $f$-harmonic $1$-forms; in particular, in dimension $2$, the procedure gives an index estimate in terms of the genus of the surface.

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arXiv2017

The Steklov spectrum of cuboids

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids.

math.SP↗
arXiv2017

Eigenvalue upper bounds for the magnetic Schroedinger operator

We study the eigenvalues of the magnetic Schroedinger operator associated with a magnetic potential A and a scalar potential q, on a compact Riemannian manifold M, with Neumann boundary conditions if the boundary is not empty. We obtain several bounds for the spectrum. Besides the dimension and the volume of the manifold, the geometric quantity which plays an important role in these estimates is the first eigenvalue of the Hodge-de Rham Laplacian acting on co-exact 1-forms. In the 2-dimensional case, this is nothing but the first positive eigenvalue of the Laplacian acting on functions. As for the dependence of the bounds on the potentials, it brings into play the mean value of the scalar potential q, the L^2-norm of the magnetic field B=dA, and the distance, taken in L^2, between the harmonic component of A and the subspace of all closed 1-forms whose cohomology class is integral (that is, having integral flux around any loop). In particular, this distance is zero when the first cohomology group is trivial.

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arXiv2017

Lower bounds for the first eigenvalue of the magnetic Laplacian

We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.

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