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Mouhamed Moustapha Fall

Publications and source records attributed to Mouhamed Moustapha Fall.

At least 37 records · Page 2Linked to original sources
arXiv2018

Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions

We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.

math.AP↗
arXiv2018

Regularity estimates for nonlocal Schrödinger equations

We prove Hölder regularity estimates up to the boundary for weak solutions $u$ to nonlocal Schrödinger equations subject to exterior Dirichlet conditions in an open set $Ω\subset \mathbb{R}^N$. The class of nonlocal operators considered here are defined, via Dirichlet forms, by kernels $K(x,y)$ bounded from above and below by $|x-y|^{N+2s}$, with $s\in (0,1)$. The entries in the equations are in some Morrey spaces and the underline domain $Ω$ satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When $K$ defines a nonlocal operator with sufficiently regular coefficients, we obtain Hölder estimates, up to the boundary of $ Ω$, for $u$ and the ratio $u/d^s$, with $d(x)=\textrm{dist}(x,\mathbb{R}^N\setminusΩ)$. If the kernel $K$ defines a nonlocal operator with Hölder continuous coefficients and the entries are Hölder continuous, we obtain interior $C^{2s+β}$ regularity estimates of the weak solutions $u$. Our argument is based on blow-up analysis and compact Sobolev embedding.

math.AP↗
arXiv2018

Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains $Ω$ of a Riemannian manifold $(\mathcal{M},g)$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $μ_2(Ω)$ with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case when $μ_2(Ω)$ is not a simple eigenvalue. In the second part of the paper, we focus on product manifolds of the form $\mathcal{M} = \mathbb{R}^k \times \mathcal{N}$, and we classify the subdomains where an associated overdetermined boundary value problem has a solution.

math.AP↗
arXiv2017

Periodic patterns for a model involving short-range and long-range interactions

We consider a physical model where the total energy is governed by surface tension and attractive screened Coulomb potential on the 3-dimensional space. We obtain different periodic equilibrium patterns i.e. stationary sets for this energy, under some volume constraints. The patterns bifurcate smoothly from straight lamellae, lattices of round solid cylinders and lattices of round balls.

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

Liouville theorems for a general class of nonlocal operators

In this paper, we study the equation $\mathcal{L} u=0$ in $\mathbb{R}^N$, where $\mathcal{L}$ belongs to a general class of nonlocal linear operators which may be anisotropic and nonsymmetric. We classify distributional solutions of this equation, thereby extending and generalizing recent Liouville type theorems in the case where $\mathcal{L}= (-Δ)^s$, $s \in (0,1)$ is the classical fractional Laplacian.

math.AP↗
arXiv2017

On isolated singularities of Kirchhoff--type Laplacian problems

In this paper, we study isolated singular positive solutions for the following Kirchhoff--type Laplacian problem: \begin{equation*} -\left(θ+\int_Ω |\nabla u| dx\right)Δu =u^p \quad{\rm in}\quad Ω\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial Ω, \end{equation*} where $p>1$, $θ\in \R$, $Ω$ is a bounded smooth domain containing the origin in $\R^N$ with $N\ge 2$. In the subcritical case: $1 0$. To estimate $M_θ(u)$, we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that $M_θ(u)<0$, by analyzing relationship between the parameter $λ$ and the unique solution $u_λ$ of $$-Δu+λu^p=kδ_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0).$$ In the supercritical case: $N/(N-2)\le p<(N+2)/(N-2)$ with $N\ge3$, we obtain two isolated singular solutions $u_i$ with $i=1,2$ such that $M_θ(u_i)>0$ under some appropriate assumptions.

math.AP↗
arXiv2017

Delaunay hypersurfaces with constant nonlocal mean curvature

We study hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in $\mathbb{R}^N$, $N\geq 2$, all of them with the same constant nonlocal mean curvature, and bifurcating from a straight cylinder. These are Delaunay type cylinders in the nonlocal setting. The proof uses the Crandall-Rabinowitz theorem applied to a quasilinear type fractional elliptic equation.

math.DG↗
arXiv2017

Near-sphere lattices with constant nonlocal mean curvature

We are concerned with unbounded sets of $\mathbb{R}^N$ whose boundary has constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We construct CNMC sets which are the countable union of a certain bounded domain and all its translations through a periodic integer lattice of dimension $M\leq N$. Our CNMC sets form a $C^2$ branch emanating from the unit ball alone and where the parameter in the branch is essentially the distance to the closest lattice point. Thus, the new translated near-balls (or near-spheres) appear from infinity. We find their exact asymptotic shape as the parameter tends to infinity.

math.AP↗
arXiv2017

Hardy-Sobolev inequality with singularity a curve

We consider a bounded domain $Ω$ of $\mathbb{R}^N$, $N\geq 3$, and $h$ a continuous function on $Ω$. Let $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u\in H^1_0(Ω)$ to the equation $$ -Δu+h u=ρ^{-σ}_Γu^{2^*_σ-1} \qquad \textrm{ in } Ω$$ where $2^*_σ:=\frac{2(N-σ)}{N-2}$, $σ\in (0,2)$, and $ρ_Γ$ is the distance function to $Γ$. For $N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-Δ+h$ is positive at a point of the curve.

math.AP↗
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↗
arXiv2015

Entire $s$-harmonic functions are affine

In this paper, we prove that solutions to the equation $(-Δ)^s u=0$ in $\mathbb{R}^N$, for $s\in (0,1)$, are affine. This will allow us to prove uniqueness of the Riesz potential $|x|^{2s-N}$ in Lebesgue spaces.

math.AP↗
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↗
arXiv2015

Ground states and concentration phenomena for the fractional Schrödinger equation

We consider here solutions of the nonlinear fractional Schrödinger equation $$ε^{2s}(-Δ)^s u+V(x)u=u^p.$$ We show that concentration points must be critical points for $V$. We also prove that, if the potential $V$ is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as $ε$ tends to zero. In addition, if the potential $V$ is radial, then the minimizer is unique provided $ε$ is small.

math.AP↗
arXiv2015

Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay

We are concerned with hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in $\mathbb{R}^N$ with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or "cylinders" in $\mathbb{R}^2$ with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay type bands in the nonlocal setting. Here we use a Lyapunov-Schmidt procedure for a quasilinear type fractional elliptic equation.

math.AP↗