arXiv subjects

Andrea Bisterzo

Publications and source records attributed to Andrea Bisterzo.

10 recordsLinked to original sources

arXiv2026

Rigidity for two-phase overdetermined problems in $\mathbb{S}^n$

In this paper, we study rigidity phenomena associated with a class of overdetermined two-phase problems on the $n$-dimensional sphere $\mathbb{S}^n$. Specifically, we analyze the solutions of coupled elliptic equations defined on a domain $Ω\subset \mathbb{S}^n$ and its complement, subject to Dirichlet boundary conditions on the interface $\partialΩ$ and a compatibility condition on the gradient. We utilize two different methods. The method of moving planes works when $\partialΩ$ is contained inside an open hemisphere. The other method works when $Ω$ is simply connected in $\mathbb{S}^2$. Both lead to the same conclusion. Namely, the existence of a solution to such overdetermined problems necessarily implies that $Ω$ is a geodesic ball. In the latter, we prove this rigidity property for three distinct scenarios: a baseline problem with piecewise constant source terms, which is the two-phase version of the torsion problem; a generalization including a class of positive and regular nonlinearities; and finally an eigenvalue problem involving the first Dirichlet eigenvalue of the respective phases.

math.AP↗

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.

math.AP↗

arXiv2025

Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary

We investigate the overdetermined torsion problem $\begin{cases} -Δu = 1 & \text{in}\ Ω\\ u=0 & \text{on}\ \partial Ω\\ \frac{\partial u}{\partial ν}=\text{const.} & \text{on}\ \partial Ω, \end{cases}$ where $Ω$ is a smooth Riemannian domain. Domains admitting a solution to this problem are called \textit{Serrin domains}, after the celebrated work of Serrin \cite{Se71}, where is proved that in $\mathbb{R}^n$ such domains are geodesic balls. In the present paper we establish the existence of two distinct types of Serrin domains of $\mathbb{S}^3$, respectively of small and large volume, each of whose boundary is connected and is neither isometric to a geodesic sphere nor to a Clifford torus. These domains arise as nontrivial perturbations of some classical symmetric solutions to the same problem. Our approach relies on an implicit construction based on the Crandall-Rabinowitz bifurcation theorem, which allows us to detect branches of non-radial solutions bifurcating from a family of radial ones. The resulting examples highlight new geometric configurations of the torsion problem in the three-dimensional sphere, providing another proof of the fact that the rigidity of Serrin-type results can fail in the presence of curvature.

math.AP↗

arXiv2025

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

Let $M$ be a Riemannian manifold and $Ω$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $Ω$, and is $0$ on its complement. Heat will then flow away from $Ω$ to its complement, and we are interested in the temperature on the boundary of $Ω$ 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.

math.DG↗

arXiv2024

Quantitative spectral stability for compact operators

This paper deals with quantitative spectral stability for compact operators acting on $L^2(X,m)$, where $(X,m)$ is a measure space. Under fairly general assumptions, we provide a characterization of the dominant term of the asymptotic expansion of the eigenvalue variation in this abstract setting. Many of the results about quantitative spectral stability available in the literature can be recovered by our analysis. Furthermore, we illustrate our result with several applications, e.g. quantitative spectral stability for a Robin to Neumann problem, conformal transformations of Riemann metrics, Dirichlet forms under the removal of sets of small capacity, and for families of pseudo-differentials operators.

math.AP↗

arXiv2023

$L^p$ positivity preservation and self-adjointness on incomplete Riemannian manifolds

The aim of this paper is to prove a qualitative property, namely the preservation of positivity, for Schrödinger-type operators acting on $L^p$ functions defined on (possibly incomplete) Riemannian manifolds. A key assumption is a control of the behaviour of the potential of the operator near the Cauchy boundary of the manifolds. As a by-product, we establish the essential self-adjointness of such operators, as well as its generalization to the case $p\neq 2$, i.e. the fact that smooth compactly supported functions are an operator core for the Schrödinger operator in $L^p$.

math.AP↗

arXiv2023

Maximum principles in unbounded Riemannian domains

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kind of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

math.AP↗

arXiv2023

$L^p_{loc}$ positivity preservation and Liouville-type theorems

On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-Δu + λu \geq 0$ with $λ>0$ a locally bounded function that may decay to $0$ at infinity. Under suitable growth conditions on the $L^{p}$ norm of $u$ over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized $L^{p}$-preservation property that can be read as a Liouville type property for nonnegative subsolutiuons of the equation $Δu \geq λu$. An application of the analytic results to $L^{p}$ growth estimates of the extrinsic distance of complete minimal submanifolds is also given.

math.AP↗

arXiv2022

Symmetry of solutions of semilinear PDEs on Riemannian domains

This paper deals with symmetry phenomena for solutions of the Dirichlet problem involving semilinear PDEs on Riemannian domains. We shall present a rather general framework where the symmetry problem can be formulated and provide some evidence that this framework is completely natural by pointing out some results for stable solutions. The case of manifolds with density, and corresponding weighted Laplacians, is inserted in the picture from the very beginning.

math.AP↗

arXiv2021

The $L^\infty$-positivity preserving property and stochastic completeness

We say that a Riemannian manifold satisfies the $L^p$-positivity preserving property if $(-Δ+ 1)u\ge 0$ in a distributional sense implies $u \ge 0$ for all $ u \in L^p$.While geodesic completeness of the manifold at hand ensures the $L^p$-positivity preserving property for all $p \in (1, +\infty)$, when $p = + \infty$ some assumptions are needed. In this paper we show that the $L^\infty$-positivity preserving property is in fact equivalent to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. The result is achieved via some monotone approximation results for distributional solutions of $-Δ+ 1 \ge 0$, which are of independent interest.

math.AP↗