arXiv subjects

Ignace Aristide Minlend

Publications and source records attributed to Ignace Aristide Minlend.

16 recordsLinked to original sources

arXiv2025

An Overdetermined Neumann boundary value problem with a general driving force

In this paper, we prove the existence of a family of non trivial compact subdomains $Ø$ in the manifold $\mathcal{M}=\R^N\times \R/2π\Z, N\geq 2$ for which the overdetermined Neumann boundary value problem \begin{align}\label{Neumann1} \left \{ \begin{aligned} $-\D w&=μg(w) && \qquad \text{in $ Ω$,}$ \frac{\partial w}{\partialη} &=0 &&\qquad \text{on $\partial Ω,$} w&=c\ne 0 &&\qquad \text{on $\partial Ω$,} \end{aligned} \right. \end{align} admits solutions for some $μ> 0$ and a $C^{1, α}$ function $g:\R \rightarrow \R.$ The domains we construct have nonconstant principal curvature, and therefore are not isoparametric nor homogeneous. The argument we develop applies for both linear and non-linear functions $g$. By this, we generalise a recent result obtained by Fall, Weth and the first named author in \cite{Fall-MinlendI-Weth4}, where the overdetermined Neumann eigenvalue problem for the Laplacian was considered.

math.AP↗

arXiv2024

The Schiffer problem on the cylinder and on the $2$-sphere

We prove the existence of a family of compact subdomains $Ω$ of the flat cylinder $\mathbb{R}^N\times \mathbb{R}/2π\mathbb{Z}$ for which the Neumann eigenvalue problem for the Laplacian on $Ω$ admits eigenfunctions with constant Dirichlet values on $\partial Ω$. These domains $Ω$ have the property that their boundaries $\partial Ω$ have nonconstant principal curvatures. In the context of ambient Riemannian manifolds, our construction provides the first examples of such domains whose boundaries are neither homogeneous nor isoparametric hypersurfaces. The functional analytic approach we develop in this paper overcomes an inherent loss of regularity of the problem in standard function spaces. With the help of this approach, we also construct a related family of subdomains of the $2$-sphere $S^2$. By this we disprove a conjecture in \cite{Souam}.

math.AP↗

arXiv2023

Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains

We prove the existence of nontrivial unbounded domains $Ø$ in the Euclidean space $\R^d$ for which the Dirichlet eigenvalue problem for the Laplacian on $Ω$ admits sign-changing eigenfunctions with constant Neumann values on $\partial Ω$. We also establish a similar result by studying a partially overdetermined problem on domains with two boundary components and opposite Neumann boundary values. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from straight (generalized) cylinder or slab.

math.AP↗

arXiv2023

Exceptional domains in higher dimensions

We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space $\R^N$, $N\geq4$. These domains arise as perturbations of complements of straight cylinders in $\R^N$, and by definition they support a positive harmonic function with vanishing Dirichlet boundary values and constant Neumann boundary values, the so-called roof function. While the domains have a similar shape as those constructed in the recent work \cite{Fall-MinlendI-Weth3} for the case $N=3$, there is a striking constrast with regard to the shape of corresponding roof functions which are bounded for $N \ge 4$. Moreover, while the analysis in \cite{Fall-MinlendI-Weth3} does not extend to higher dimensions, the approach of the present paper depends heavily on the assumption $N \ge 4$.

math.AP↗

arXiv2022

On an electrostatic problem and a new class of exceptional subdomains of $\mathbb{R}^3$

We study the existence of nontrivial unbounded surfaces $S\subset \mathbb{R}^3$ with the property that the constant charge distribution on $S$ is an electrostatic equilibrium, i.e. the resulting electrostatic force is normal to the surface at each point on $S$. Among bounded regular surfaces $S$, only the round sphere has this property by a result of Reichel $[23]$ (see also Mendez and Reichel $[16]$) confirming a conjecture of P. Gruber. In the present paper, we show the existence of nontrivial exceptional domains $Ω\subset \mathbb{R}^3$ whose boundaries $S=\partial Ω$ enjoy the above property.

math.AP↗

arXiv2022

An overdetermined problem for sign-changing eigenfunctions in unbounded domains

We study the existence of non-trivial unbounded domains of $Ω\subset \mathbb{R}^2$ where the equation \begin{align} - λu_{xx} -u_{tt} &= u \qquad \text{in $Ω$,}\nonumber u &=0 \qquad \text{on $\partial Ω$,}\nonumber \end{align} is solvable subject to the conditions \begin{align} \frac{\partial u}{\partial η} =-1\quad \text{on $\partial Ω^+$} \quad \textrm{and}\quad \frac{\partial u}{\partial η} =+1\quad \text{on $\partial Ω^-$.} \end{align} For every integer $m\geq 0$, we prove the existence of a family of unbounded domains $Ω\subset \mathbb{R}^2$ indexed by $0 \leqslant\ell\leqslant 2m$, where the above problem admits periodic sign-changing solutions. The domains we construct are periodic in the first coordinate in $\mathbb{R}^2$, and they bifurcate from suitable strips.

math.AP↗

arXiv2021

Unbounded periodic constant mean curvature graphs on Calibrable Cheeger Serrin domains

We prove a general result characterizing a specific class of Serrin domains as supports of unbounded and periodic constant mean curvature graphs. We apply this result to prove the existence of a family of unbounded periodic constant mean curvature graphs, each supported by a Serrin domain and intersecting its boundary orthogonally, up to a translation. We also show that the underlying Serrin domains are calibrable and Cheeger in a suitable sens, and they solve the 1-Laplacian equation.

math.AP↗

arXiv2020

Overdetermined problems for fully nonlinear equations in space forms

We study overdetermined problems for fully nonlinear elliptic equations in subdomains $Ø$ of the Euclidean sphere $\mathbb{S}^{N}$ and the hyperbolic space $\mathbb{H}^{N}$. We prove, the existence of a classical solution to the underlined equation forces $Ø$ to be a geodesic ball in the ambient space. Our result extends to fully nonlinear equations, a similar result in the case of semilinear equations with the Laplace operator due to Kumaresan and Prajapat.

math.AP↗

arXiv2020

Foliation of an asymptotically flat end by critical capacitors

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace-Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the non-local nature of our problem

math.AP↗

arXiv2018

Multiply-periodic hypersurfaces with constant nonlocal mean curvature

We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of smooth branches of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.

math.AP↗

arXiv2017

Serrin's overdetermined problem on the sphere

We study Serrin's overdetermined boundary value problem \begin{equation*} -Δ_{S^N}\, u=1 \quad \text{ in $Ω$},\qquad u=0, \; \partial_ηu=\textrm{const} \quad \text{on $\partial Ω$} \end{equation*} in subdomains $Ω$ of the round unit sphere $S^N \subset \mathbb{R}^{N+1}$, where $Δ_{S^N}$ denotes the Laplace-Beltrami operator on $S^N$. A subdomain $Ω$ of $S^N$ is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in $S^{N}$ which are not bounded by geodesic spheres.

math.AP↗

arXiv2017

Existence of Self-Cheeger sets on Riemannian manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold of dimension $N\geq 2$. We prove the existence of a family $(Ω_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ of self-Cheeger sets in $(\mathcal{M},g)$ . The domains $Ω_\varepsilon\subset\mathcal{M}$ are perturbations of geodesic balls of radius $\varepsilon$ centered at $p \in \mathcal{M}$, and in particular, if $p_0$ is a non-degenerate critical point of the scalar curvature of $g$, then the family $(\partial Ω_\varepsilon)_{\varepsilon \in (0,\varepsilon_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$.

math.DG↗

arXiv2016

Unbounded periodic solutions to Serrin's overdetermined boundary value problem

We study the existence of nontrivial unbounded domains $Ω$ in $\mathbb{R}^N$ such that the overdetermined problem $$ -Δu = 1 \quad \text{in $Ω$}, \qquad u=0, \quad \partial_νu=\textrm{const} \qquad \text{on $\partial Ω$} $$ admits a solution $u$. By this, we complement Serrin's classification result from 1971 which yields that every bounded domain admitting a solution of the above problem is a ball in $\mathbb{R}^N$. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the problem.

math.AP↗

arXiv2016

Existence of Self-Cheeger Sets on Riemannian Manifolds

Let $(\mathcal{M}, g)$ be a compact Riemannian manifold of dimension $N\geq 2$. We prove the existence of a family $(Ω_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ of self-Cheeger sets in $(\mathcal{M}, g)$ . The domains $Ω_\varepsilon\subset\mathcal{M}$ are perturbations of geodesic balls of radius $\varepsilon$ centered at $p \in \mathcal{M}$, and in particular, if $p_0$ is a non-degenerate critical point of the scalar curvature of $g$, then the family $( \partialΩ_\varepsilon)_{\varepsilon\in (0,\varepsilon_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$.

math.DG↗

arXiv2015

The role of the mean curvature in a Hardy-Sobolev trace inequality

The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

math.AP↗

arXiv2014

Serrin's over-determined Problem on Riemannian Manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold of dimension $N$, $N\geq 2$. In this paper, we prove that there exists a family of domains $(Ω_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ and functions $u_\varepsilon$ such that $ -Δ_{g} u_\varepsilon=1 \quad \textrm{ in } Ω_\varepsilon, \quad u_\varepsilon=0 \quad\textrm{ on }\partialΩ_\varepsilon, \quad {g}(\nabla_{ {g}} {u_\varepsilon}, ν_\varepsilon)=-\frac{\varepsilon}{N} \quad \textrm{ on }\partialΩ_\varepsilon, $ where $ν_\varepsilon$ is the unit outer normal of $\partialΩ_\varepsilon$. The domains $Ω_\varepsilon$ are smooth perturbations of geodesic balls of radius $\varepsilon$ centered at some point $p_0$. If, in addition, $p_0$ is a non-degenerate critical point of the scalar curvature of $g$ then, the family $(\partialΩ_\varepsilon)_{\varepsilon\in(0,\varepsilon_0)}$ constitutes a smooth foliation of a neighborhood of $p_0$. By considering a family of domains $Ω_\varepsilon$ in which the above overdetermined system is satisfied, we also prove that if this family converges to some point $p_0$ in a suitable sense as $\varepsilon\to 0$, then $p_0$ is a critical point of the scalar curvature. A Taylor expansion of he energy rigidity for the torsion problem is also given.

math.DG↗