arXiv subjects

Mouhamed Moustapha Fall

Publications and source records attributed to Mouhamed Moustapha Fall.

At least 19 recordsLinked to original sources

arXiv2026

Delaunay-type interface in a screened model of diblock copolymer melts

A diblock copolymer is a soft-matter composed of two chemically distinct block of repeating monomers covalently bonded together at an end-to-end junction to form a single polymer chain. In this paper, we establish the existence of infinitely many smooth periodic unbounded domain patterns of Delaunay-type in $\mathbb{R}^3$ that optimize the energy distribution in diblock copolymer melts. We emphasize that pattern domains at the equilibrium correspond to stationary sets of the screened Ohta--Kawasaki free energy functional \begin{align*} \mathcal{P}_γ(Ω) := |\partialΩ| + γ\int_Ω\int_Ω G_κ(|x-y|) \,\mathrm{d}x\mathrm{d}y, \end{align*} where $γ>0$, $κ>0$ and $G_κ(r)=\frac{1}{r} e^{-κr}$ is the repulisive Yukawa potential. Equivalently, these equilibria satisfy the corresponding Euler--Lagrange equation \begin{align*} \mathcal{H}_Ω(x):= H_{\partialΩ}(x) + γ\int_Ω G_κ(|x-y|) \mathrm{d}y = \textrm{Const} \quad \text{on } \partialΩ, \end{align*} where $H_{\partialΩ}$ denotes the mean curvature of the surface $\partialΩ$. By analyzing the linearization of $Ω\mapsto \mathcal{H}_Ω$ around flat cylinders and applying the Crandall--Rabinowitz bifurcation theorem, for any $κ> 0$ and sufficiently small $γ> 0$, we prove the existence of non-trivial, $2π$-periodic Delaunay-type equilibrium cylinder interfaces with shapes close to a Delaunay unduloid surface of constant mean curvature.

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arXiv2025

Second radial eigenfunctions to a fractional Dirichlet problem and uniqueness for a semilinear equation

We analyze the shape of radial second Dirichlet eigenfunctions of fractional Schrödinger type operators of the form $(-Δ)^s +V$ in the unit ball $B$ in $\mathbb{R}^N$ with a nondecreasing radial potential $V$. Specifically, we show that the eigenspace corresponding to the second radial eigenvalue is simple and spanned by an eigenfunction $u$ which changes sign precisely once in the radial variable and does not have zeroes anywhere else in $B$. Moreover, by a new Hopf type lemma for supersolutions to a class of degenerate mixed boundary value problems, we show that $u$ has a nonvanishing fractional boundary derivative on $\partial B$. We apply this result to prove uniqueness and nondegeneracy of positive ground state solutions to the problem $(-Δ)^s u+λu=u^p$ on ${B}$, $\; u=0$ on $\mathbb{R}^N\setminus B$. Here $s\in (0,1)$, $λ\geq 0$ and $p>1$ is strictly smaller than the critical Sobolev exponent.

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arXiv2025

Overdetermined problems with fractional Laplacian

Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $Ω$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -Δ)^s u = f(u) \text{ in $Ω$,} \qquad u=0 \text{ in $\mathbb{R}^N\setminus Ω$,} \qquad(\partial_η)_s u=Const. \text{ on $\partial Ω$} \end{equation*} has a nonnegative and nontrivial solution, where $η$ is the outer unit normal vectorfield along $\partialΩ$ and for $x_0\in\partialΩ$ \[ \left(\partial_η\right)_{s}u(x_{0})=-\lim_{t\to 0}\frac{u(x_{0}-tη(x_0))}{t^s}. \] Under mild assumptions on $f$, we prove that $Ω$ must be a ball. In the special case $f\equiv 1$, we obtain an extension of Serrin's result in 1971. The fact that $Ω$ is not assumed to be connected is related to the nonlocal property of the fractional Laplacian. The main ingredients in our proof are maximum principles and the method of moving planes.

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arXiv2024

Nondegeneracy properties and uniqueness of positive solutions to a class of fractional semilinear equations

We prove that positive solutions $u\in H^s(\mathbb{R}^N)$ to the equation $(-Δ)^s u+ u=u^p$ in $\mathbb{R}^N$ are nonradially nondegenerate, for all $s\in (0,1)$, $N\geq 1$ and $p>1$ strictly smaller than the critical Sobolev exponent. By this we mean that the linearized equation $(-Δ)^s w+ w-pu^{p-1}w = 0$ does not admit nonradial solutions beside the directional derivatives of $u$. Letting $B$ be the unit centered ball and $λ_1(B)$ the first Dirichlet eigenvalue of the fractional Laplacian $(-Δ)^s$, we also prove that positive solutions to $(-Δ)^s u+λu=u^p$ in ${B}$ with $u=0$ on $\mathbb{R}^N\setminus B$, are nonradially nondegenerate for any $λ> -λ_1(B)$ in the sense that the linearized equation does not admit nonradial solutions. From these results, we then deduce uniqueness and full nondegeneracy of positive solutions in some special cases. In particular, in the case $N=1$, we prove that the equation $(-Δ)^s u+ u=u^2$ in $\mathbb{R}$ or in $B$, with zero exterior data, admits a unique even solution which is fully nondegenerate in the optimal range $s \in (\frac{1}{6},1)$, thus extending the classical uniqueness result of Amick and Toland on the Benjamin-Ono equation. Moreover, in the case $N=1$, $λ=0$, we also prove the uniqueness and full nondegeneracy of positive solutions for the Dirichlet problem in $B$ with arbitrary subcritical exponent $p$. Finally, we determine the unique positive ground state solution of $(-Δ)^{\frac{1}{2}} u+ u=u^{p}$ in $\mathbb{R}^N$, $N \ge 1$ with $p=1+\frac{2}{N+1}$ and compute the sharp constant in the associated Gagliardo-Nirenberg inequality $$ \|u\|_{L^{p+1}(\mathbb{R}^N)} \le C \|(-Δ)^{\frac{1}{4}} u\|_{L^2(\mathbb{R}^N)}^{\frac{N}{N+2}} \|u\|_{L^2(\mathbb{R}^N)}^{\frac{2}{N+2}}. $$

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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}.

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arXiv2023

Generic properties of eigenvalues of the fractional Laplacian

We consider the Dirichlet eigenvalues of the fractional Laplacian $(-Δ)^s$, with $s\in (0,1)$, related to a smooth bounded domain $Ω$. We prove that there exists an arbitrarily small perturbation $\tildeΩ=(I+ψ)(Ω)$ of the original domain such that all Dirichlet eigenvalues of the fractional Laplacian associated to $\tildeΩ$ are simple. As a consequence we obtain that all Dirichlet eigenvalues of the fractional Laplacian on an interval are simple. In addition, we prove that for a generic choice of parameters all the eigenvalues of some non-local operators are also simple.

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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.

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arXiv2022

A Hopf lemma for the regional fractional Laplacian

We provide a Hopf boundary lemma for the regional fractional Laplacian $(-Δ)^s_Ω$, with $Ω\subset\mathbb{R}^N$ a bounded open set. More precisely, given $u$ a pointwise or weak super-solution of the equation $(-Δ)^s_Ω u = c(x)u$ in $Ω$, we show that the ratio $u(x)/(\mathrm{dist}(x,\partialΩ))^{2s-1}$ is strictly positive as $x$ approaches the boundary $\partialΩ$ of $Ω$. We also prove a strong maximum principle for distributional super-solutions.

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arXiv2022

A fractional Hadamard formula and applications

We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ λ_{s,p}(Ω):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(Ω),\,\, \|u\|_{L^p(Ω)}=1 \right\}, $$ where $s\in (0,1)$, $Ω$ is bounded of class $C^{1,1}$ and $p\in [1, \frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we derive formula for the one-sided shape derivative of the mapping $Ω\mapsto λ_{s,p}(Ω)$ under domain perturbations. In the case where $ λ_{s,p}(Ω)$ admits a unique positive minimizer (e.g. $p=1$ or $p=2$), our result implies a nonlocal version of the classical variational Hadamard formula for the first eigenvalue of the Dirichlet Laplacian on $Ω$. Thanks to the formula for our one-sided shape derivative, we characterize smooth local minimizers of $λ_{s,p}(Ω)$ under volume-preserving deformations, and we find that they are balls if $p\in \{1\}\cup [2,\infty)$. Finally, we consider the maximization problem for $λ_{s,p}(Ω)$ among annular-shaped domains of fixed volume of the type $B\setminus \overline B'$, where $B$ is a fixed ball and $B'$ is ball whose position is varied within $B$. We prove that, for $p\in \{1,2\}$, the value $λ_{s,p}(B\setminus \overline B')$ is maximal when the two balls are concentric.

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arXiv2022

Regional fractional Laplacians: Boundary regularity

We study boundary regularity for solutions to a class of equations involving the so called regional fractional Lapacians $(-Δ)^s_Ω$, with $Ω\subset \mathbb{R}^N$. Recall that the regional fractional Laplacians are generated by symmetric stable processes which are not allowed to jump outside $Ω$. We consider weak solutions to the equation $(-Δ)^s_Ωw(x)=p.v.\int_Ω\frac{w(x)-w(y)}{|x-y|^{N+2s}}\, dy=f(x)$, for $s\in (0,1)$, subject to zero Neumann or Dirichlet boundary conditions. The boundary conditions are defined by considering $w$ as well as the test functions in the fractional Sobolev spaces $H^s(Ω)$ or $H^s_0(Ω)$ respectively. While the interior regularity is well understood for these problems, little is known in the boundary regularity, mainly for the Neumann problem. Under optimal regularity assumptions on $Ω$ and provided $f\in L^p(Ω)$, we show that $w\in C^{2s-N/p}(\overline Ω)$ in the case of zero Neumann boundary conditions. As a consequence for $2s-N/p>1$, $w\in C^{1,2s-\frac{N}{p}-1}(\overlineΩ)$. As what concerned the Dirichlet problem, we obtain ${w}/{δ^{2s-1}}\in C^{1-N/p}(\overlineΩ)$, provided $p>N$ and $s\in (1/2,1)$, where $δ(x)=\textrm{dist}(x,\partialΩ)$. To prove these results, we first classify all solutions having a certain growth at infinity when $Ω$ is a half-space and the right hand side is zero. We then carry over a fine blow up and some compactness arguments to get the results.

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arXiv2022

Existence results for nonlocal problems governed by the regional fractional Laplacian

The aim of the present paper is to study existence results of minimizers of the critical fractional Sobolev constant on bounded domains. Under some values of the fractional parameter we show that the best constant is achieved. If moreover the underlying domain is a ball, we obtain positive radial minimizers for all possible values of the fractional parameter in higher dimension, while we impose a positive mass condition in low dimension.

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arXiv2021

Calderon-Zygmund theory for non-convolution type nonlocal equations with continuous coefficient

Given $2\leq p<\infty$, $s\in (0, 1)$ and $t\in (1, 2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form \[ \int_Ω \int_Ω K\left (x,|x-y|,\frac{x-y}{|x-y|}\right ) \frac{(u(x)-u(y))(φ(x)-φ(y))}{|x-y|^{n+2s}} dx dy = g[φ], \quad \forall ϕ\in C_c^{\infty}(Ω) \] where $Ω\subset \mathbb{R}^{n}$ is an open set. Here we assume $K$ is bounded, nonnegative and continuous in the first entry -- and ellipticity is ensured by assuming that $K$ is strictly positive in a cone. The setup is chosen so that it is applicable for nonlocal equations on manifolds, but the structure of the equation is general enough that it also applies to the certain fractional $p$-Laplace equations around points where $u \in C^1$ and $|\nabla u| \neq 0$.

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arXiv2021

Global Schauder theory for minimizers of the $H^s(Ω)$ energy

We study the regularity of minimizers of the functional $\mathcal E(u):= [u]_{H^s(Ω)}^2 +\int_Ωfu$. This corresponds to understanding solutions for the regional fractional Laplacian in $Ω\subset\mathbb R^N$. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in $H^s(Ω)$ (i.e., Neumann problem), or in the case of Dirichlet condition $u\in H^s_0(Ω)$ when $s>\frac12$. Our main result establishes the sharp regularity of solutions in both cases: $u\in C^{2s+α}(\overlineΩ)$ in the Neumann case, and $u/δ^{2s-1}\in C^{1+α}(\overlineΩ)$ in the Dirichlet case. Here, $δ$ is the distance to $\partialΩ$, and $α<α_s$, with $α_s\in (0,1-s)$ and $2s+α_s>1$. We also show the optimality of our result: these estimates fail for $α>α_s$, even when $f$ and $\partialΩ$ are $C^\infty$.

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arXiv2021

Morse index versus radial symmetry for fractional Dirichlet problems

In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-Δ)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet eigenvalue problem for the operator $(-Δ)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Bañuelos and Kulczycki.

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arXiv2021

Nonlocal diffusion of smooth sets

We consider normal velocity of smooth sets evolving by the $s-$fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for $s\in [\frac{1}{2}, 1)$ while, for $s\in (0, \frac{1}{2})$, it is nearly proportional to the fractional mean curvature of the initial set. Our results show that the motion by (fractional) mean curvature flow can be approximated by fractional heat diffusion and by a diffusion by means of harmonic extension of smooth sets.

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arXiv2020

Regularity results for nonlocal equations and applications

We introduce the concept of $C^{m,α}$-nonlocal operators, extending the notion of second order elliptic operator in divergence form with $C^{m,α}$-coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the Hölder continuity of the gradient of the solutions in the case of $C^{0,α}$-coefficients and the classical Shauder estimates for $C^{m+1,α}$-coefficients. We further apply the regularity results for $C^{m,α}$-nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided.

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arXiv2019

Gradient estimates in fractional Dirichlet problems

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.

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arXiv2018

Constant Nonlocal Mean Curvatures surfaces and related problems

The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear differential operators that are involved when it acts on graphs. We also survey recent results on surfaces having constant NMC and describe their intimate link with some problems arising in the study of overdetermined boundary value problems.

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