arXiv subjects

Pieralberto Sicbaldi

Publications and source records attributed to Pieralberto Sicbaldi.

At least 19 recordsLinked to original sources

arXiv2026

Bifurcation of overdetermined capillary problems in a strip domain

In this paper, we consider the classical overdetermined capillary problem: \begin{equation*} \begin{cases} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) - bu =0 &~~\mbox{in}~~ \Omega, \partial_{\nu} u=\kappa &~~\mbox{on}~~\partial\Omega, u=c &~~\mbox{on}~~\partial\Omega, \end{cases} \end{equation*} where $b$, $c$ and $\kappa$ are positive constants, and $\Omega\subset \mathbb{R}^2$. When $\Omega$ is an infinite strip, i.e., a domain bounded by two parallel straight lines, there exists a unique one-dimensional solution (called the trivial solution) to this problem. By means of a bifurcation argument, we establish the existence of a critical period $T_*$ at which a branch of non-trivial solutions bifurcates from the trivial one. These solutions are genuinely two-dimensional and are defined in unbounded periodic domains $\Omega$ that are diffeomorphic to an infinite strip, yet whose boundaries are no longer straight lines. This result offers a significant physical interpretation in the context of capillary phenomena.

math.AP↗

arXiv2026

Overdetermined problems for the rotationally invariant Poisson equation in model manifolds

We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation $-\Delta_{g_\mathcal{M}} u = f(r)$ in a model manifold $\mathcal{M} = [0,S) \times_h \mathbb S^{N-1}$ with warping function $h$. The variable $r$ ranges in the interval $[0,S)$, whose endpoint $S$ is positive and possibly infinite. The first part of the paper deals with the problem \[ \begin{array}{ll} -\Delta_{g_\mathcal{M}} {u}=f(r) &\mbox{in $\Omega$}, u=\varphi(r) &\mbox{on $\partial \Omega$}, \frac{\partial u}{\partial \nu} = \kappa(r) &\mbox{on $\partial \Omega$}, \end{array} \] where $\Omega \subset \mathcal{M}$ is a bounded domain containing the point $O \in \mathcal{M}$ corresponding to $r = 0$, $\nu$ is the exterior unit normal vector on $\partial \Omega$, and $f$, $\varphi$, $\kappa$ are three prescribed functions. In the second part of the paper, we consider a similar overdetermined problem for the exterior Bernoulli problem in a domain $\Omega \setminus \overline B_{R_0}(O)$, where $B_{R_0}(O)$ denotes the geodesic ball centered at $O$ with radius $R_0$, within the class of functions that vanish on $\partial B_{R_0}(O)$. In both cases, we give conditions on $f$, $\varphi$ and $\kappa$ implying that the solution $u$ is radial and $\Omega$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms $\mathbb{R}^N$, $\mathbb{H}^N$ and $\mathbb{S}^N$.

math.AP↗

arXiv2025

Bifurcating domains for an overdetermined eigenvalue problem in cylinders

We study an overdetermined eigenvalue problem for domains $\Omega$ contained in the half-cylinder $\Sigma=\omega \times (0, +\infty)$, based on a bounded regular domain $\omega \subset \mathbb{R}^{N-1}$. It is easy to see that in any bounded cylinder $\Omega_{t}=\omega \times (0, t)$, $t > 0$, the eigenvalue problem admits a one-dimensional positive eigenfunction which satisfies the overdetermined boundary conditions. The aim of the paper is to construct other domains $\Omega\subset \Sigma$ for which there exists a positive eigenfunction that is a solution of the overdetermined problem. This is achieved by showing that branches of such domains bifurcate from the ``trivial'' domains $\Omega_{t_j}$ at the values $t_{j} = \frac{\pi}{2\sqrt{\sigma_j}}$ where $\sigma_j$ ($j\geq 1$) is a simple Neumann eigenvalue of the Laplace operator on $\omega \subset \mathbb{R}^{N-1}$. The solutions can be reflected with respect to $\omega$ to generate nontrivial solutions in a cylinder.

math.AP↗

arXiv2025

Rigidity results for the capillary overdetermined problem

In this paper we obtain rigidity results for bounded positive solutions of the general capillary overdetermined problem \begin{equation} \left\{ \begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0 & \mbox{in }\; Ω,\\[1mm] u= 0 & \mbox{on }\; \partial Ω,\\[1mm] \partial_ν u=κ&\mbox{on }\; \partial Ω, \end{array}\right. \end{equation} where $f$ is a given $C^1$ function in $\mathbb{R}$, $ν$ is the exterior unit normal, $κ$ is a constant and $Ω\subset \mathbb{R}^n$ is a $C^1$ domain. Our main theorem states that if $n=2, κ\neq 0$, $\partial Ω$ is unbounded and connected, $|\nabla u|$ is bounded and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\geq \left(1+κ^2\right)^{-\frac12} -1$, then $Ω$ must be a half-plane and $u$ is a parallel solution. In other words, under our assumptions, if a capillary graph has the property that its mean curvature depends only on the height, then it is the graph of a one dimensional function. We also prove the boundedness of the gradient of solutions of the above problem when $f'(u) <0$. Moreover we study a Modica type estimate for the above overdetermined problem that allows us to prove that, unless $Ω$ is a half-space, the mean curvature of $\partial Ω$ is strictly negative under the assumption that $κ\neq 0$ and there exists a nonpositive primitive $F$ of $f$ such that $F(0)\geq \left(1+κ^2\right)^{-\frac12} -1$. Our results have an interesting physical application to the classical capillary overdetermined problem, i.e., the case where $f$ is linear.

math.AP↗

arXiv2024

Modica type estimates and curvature results for overdetermined elliptic problems

In this paper, we establish a Modica type estimate on bounded solutions to the overdetermined elliptic problem \begin{equation*} \begin{cases} Δu+f(u) =0& \mbox{in $Ω$, }\\ u>0 &\mbox{in $Ω$, } u=0 &\mbox{on $\partialΩ$, } \partial_ν u=-κ&\mbox{on $\partialΩ$, } \end{cases} \end{equation*} where $Ω\subset\mathbb{R}^{n},n\geq 2$. As we will see, the presence of the boundary changes the usual form of the Modica estimate for entire solutions. We will also discuss the equality case. From such estimates we will deduce information about the curvature of $\partial Ω$ under a certain condition on $κ$ and $f$. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.

math.AP↗

arXiv2024

A Schiffer-type problem for annuli with applications to stationary planar Euler flows

If on a smooth bounded domain $Ω\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $Ω$ be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture, in that the function $u$ is allowed to take a different constant value on each connected component of $\partial Ω$ yet many of the known rigidity properties of the original problem are essentially preserved. Our main result provides a negative answer by constructing a family of nontrivial doubly connected domains $Ω$ with the above property. As a consequence, a certain linear combination of the indicator functions of the domains $Ω$ and of the bounded component of the complement $\mathbb{R}^2\backslash\overlineΩ$ fails to have the Pompeiu property. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial.

math.AP↗

arXiv2024

Overdetermined elliptic problems in nontrivial exterior domains of the hyperbolic space

We construct nontrivial unbounded domains $Ω$ in the hyperbolic space $\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball, such that the overdetermined elliptic problem \begin{equation} -Δ_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,Ω, \,\, u=0,\,\,\partial_νu=\text{const}\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a positive bounded solution in $C^{2,α}\left(Ω\right) \cap H^1\left(Ω\right)$. We also give a condition under which this construction holds for larger dimensions $N$. This is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems, and, as far as we know, is the first nontrivial example of solution to an overdetermined elliptic problem in the hyperbolic space.

math.AP↗

arXiv2024

Nontrivial solutions to the relative overdetermined torsion problem in a cylinder

Given a bounded regular domain $ω\subset \mathbb{R}^{N-1}$ and the half-cylinder $Σ= ω\times (0,+\infty)$, we consider the relative overdetermined torsion problem in $Σ$, i.e. \[\left\{ \begin{array}{ll} Δ{u}+1=0 &\mbox{in $Ω$},\newline \partial_ηu = 0 &\mbox{on $\widetilde Γ_Ω$},\newline u=0 &\mbox{on $Γ_Ω$},\newline \partial_νu =c &\mbox{on $Γ_Ω$}. \end{array} \right. \] where $Ω\subset Σ$, $Γ_Ω= \partial Ω\cap Σ$, $\widetilde Γ_Ω= \partial Ω\setminus Γ_Ω$, $ν$ is the outer unit normal vector on $Γ_Ω$ and $η$ is the outer unit normal vector on $\widetilde Γ_Ω$. We build nontrivial solutions to this problem in domains $Ω$ that are the hypograph of certain nonconstant functions $v : \overlineω \to (0, + \infty)$. Such solutions can be reflected with respect to $ω$, giving nontrivial solutions to the relative overdetermined torsion problem in a cylinder. The proof uses a local bifurcation argument which, quite remarkably, works for any generic base $ω$.

math.AP↗

arXiv2023

Overdetermined elliptic problems in nontrivial contractible domains of the sphere

In this paper, we prove the existence of nontrivial contractible domains $Ω\subset\mathbb{S}^{d}$, $d\geq2$, such that the overdetermined elliptic problem \begin{equation*} \begin{cases} -\varepsilonΔ_{g} u +u-u^{p}=0 &\mbox{in $Ω$, } u>0 &\mbox{in $Ω$, } u=0 &\mbox{on $\partialΩ$, } \partial_ν u=\mbox{constant} &\mbox{on $\partialΩ$, } \end{cases} \end{equation*} admits a positive solution. Here $Δ_{g}$ is the Laplace-Beltrami operator in the unit sphere $\mathbb{S}^{d}$ with respect to the canonical round metric $g$, $\varepsilon>0$ is a small real parameter and $1 1$ if $d=2$). These domains are perturbations of $\mathbb{S}^{d}\setminus D,$ where $D$ is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.

math.AP↗

arXiv2019

Half space theorem for the Allen-Cahn equation and related problems

In this paper we obtain rigidity results for a bounded non-constant entire solution $u$ of the Allen-Cahn equation in $\mathbb{R}^n$, whose level set $\{u=0\}$ is contained in a half-space. If $n\leq 3$ we prove that the solution must be one-dimensional. In dimension $n\geq 4$, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

math.AP↗

arXiv2019

Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

In this paper, we study the minimization of $λ_{1}(Ω)$, the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets $Ω$ of fixed volume in a Riemmanian manifold $(M,g)$. In the Euclidian setting (when $(M,g)=(\mathbb{R}^n,e)$), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds $(M,g)$. In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.

math.AP↗

arXiv2016

Solutions to overdetermined elliptic problems in nontrivial exterior domains

In this paper we construct nontrivial exterior domains $Ω\subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -Δu +u -u^p=0,\ u >0 & \mbox{in }\; Ω, {1mm] \ u= 0 & \mbox{on }\; \partial Ω, [1mm] \ \frac{\partial u}{\partial ν} = \mbox{cte} & \mbox{on }\; \partial Ω, \right.$$ admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found in the literature; however, our example is the first one in the form of an exterior domain.

math.AP↗

arXiv2015

A rigidity result for overdetermined elliptic problems in the plane

Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $Ω\subset \mathbb{R}^2$ a $C^{1,α}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem $$ \left\{\begin{array} {ll} Δu + f(u) = 0 & \mbox{in }\; Ω \\ u= 0\, \, \, , \, \, \, \frac{\partial u}{\partial \vecν}=1 &\mbox{on }\; \partial Ω\end{array}\right. $$ we prove that $Ω$ is a half-plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997.

math.AP↗

arXiv2014

New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.

math.DG↗

arXiv2013

Geometry and Topology of some overdetermined elliptic problems

We study necessary conditions on the geometry and the topology of domains in $\mathbb{R}^2$ that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use come from constant mean curvature surface theory. In particular, we obtain a partial answer to a question posed by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997. We investigate also some boundedness properties of the solution $u$. Some of our results generalize to higher dimensions.

math.AP↗

arXiv2013

Delaunay type domains for an overdetermined elliptic problem in S^n x R and H^n x R

We prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. The domains Ω_j are rotationally symmetric and periodic with respect to the R-axis of the cylinder and as j converges to 0 the domain Ω_j converges to the cylinder B^n x R.

math.DG↗

arXiv2013

Extremal domains for the first eigenvalue in a general Riemannian manifold

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the Riemannian manifold was required.

math.DG↗

arXiv2011

Bifurcating extremal domains for the first eigenvalue of the Laplacian

We prove the existence of a smooth family of non-compact domains $Omega_s \subset R^{n+1}$ bifurcating from the straight cylinder $B^n \times R$ for which the first eigenfunction of the Laplacian with 0 Dirichlet boundary condition also has constant Neumann data at the boundary. The domains $Omega_s$ are rotationally symmetric and periodic with respect to the R-axis of the cylinder; they are of the form $Omega_s = {(x,t) \in R^n \times R \mid |x| < 1+s \cos((2π)/T_s t) + O(s^2)}$ where $T_s = T_0 + O(s)$ and T_0 is a positive real number depending on n. For $n \ge 2$ these domains provide a smooth family of counter-examples to a conjecture of Berestycki, Caffarelli and Nirenberg. We also give rather precise upper and lower bounds for the bifurcation period T_0. This work improves a recent result of the second author.

math.DG↗