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Mouhamed Moustapha Fall

Publications and source records attributed to Mouhamed Moustapha Fall.

54 records · Page 3Linked to original sources
arXiv2015

Profile expansion for the first nontrivial Steklov eigenvalue in Riemannian manifolds

We study the problem of maximizing the first nontrivial Steklov eigenvalue of the Laplace-Beltrami Operator among subdomains of fixed volume of a Riemannian manifold. More precisely, we study the expansion of the corresponding profile of this isoperimetric (or isochoric) problem as the volume tends to zero. The main difficulty encountered in our study is the lack of existence results for maximizing domains and the possible degeneracy of the first nontrivial Steklov eigenvalue, which makes it difficult to tackle the problem with domain variation techniques. As a corollary of our results, we deduce local comparison principles for the profile in terms of the scalar curvature on $\mathcal{M}$. In the case where the underlying manifold is a closed surface, we obtain a global expansion and thus a global comparison principle.

math.AP↗
arXiv2014

Positive solutions to some nonlinear fractional Schrödinger equations via a min-max procedure

The existence of a positive solution to the following fractional semilinear equation is proven, in a situation where a ground state solution may not exist. More precisely, we consider for $0 0$, $Q\geq 0$ are bounded functions converging to $1$ as $|x|\to\infty$. Using a min-max procedure introduced by Bahri and Li we prove the existence of a positive solution under one-sided asymptotic bounds for $V$ and $Q$.

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↗
arXiv2013

Monotonicity and nonexistence results for some fractional elliptic problems in the half space

We study a class of fractional elliptic problems of the form $\Ds u= f(u)$ in the half space $\R^N_+:=\{x \in \R^N\::\: x_1>0\}$ with the complementary Dirichlet condition $u \equiv 0$ in $\R^N \setminus \R^N_+$. Under mild assumptions on the nonlinearity $f$, we show that bounded positive solutions are increasing in $x_1$. For the special case $f(u)=u^q$, we deduce nonexistence of positive bounded solutions in the case where $q \ge 1$ and $q<\frac{N-1+2s}{N-1-2s}$ if $N \ge 1+2s$. We do not require integrability assumptions on the solutions we study.

math.AP↗
arXiv2013

Sharp local estimates for the Szegö-Weinberger profile in Riemannian manifolds

We study the local Szegö-Weinberger profile in a geodesic ball $B_g(y_0,r_0)$ centered at a point $y_0$ in a Riemannian manifold $(\M,g)$. This profile is obtained by maximizing the first nontrivial Neumann eigenvalue $μ_2$ of the Laplace-Beltrami Operator $Δ_g$ on $\M$ among subdomains of $B_g(y_0,r_0)$ with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of $\M$ at $y_0$. As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of $Δ_g$, but additional difficulties arise due to the fact that $μ_2$ is degenerate in the unit ball in $\R^N$ and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.

math.DG↗
arXiv2013

Unique continuation property and local asymptotics of solutions to fractional elliptic equations

Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.

math.AP↗
arXiv2012

Weighted Hardy inequality with higher dimensional singularity on the boundary

Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.

math.AP↗
arXiv2012

Nonexistence results for a class of fractional elliptic boundary value problems

In this paper we study a class of fractional elliptic problems of the form $$ \Ds u= f(x,u) \quad \textrm{in} Øu=0\quad \textrm{in} \R^N \setminus Ø,$$ where $s\in(0,1)$. We prove nonexistence of positive solutions when $Ø$ is star-shaped and $f$ is supercritical. We also derive a nonexistence result for subcritical $f$ in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the Caffarelli-Silvestre extension \cite{CSilv} of a solution of the above problem. The standard approach in the case $s=1$ using Pohozaev type identities does not carry over to the case $0<s<1$ due to the lack of boundary regularity of solutions.

math.AP↗
arXiv2012

An extension theorem for some pseudo-differential operators

In this paper we provide an extension theorem for fractional powers of some pseudo-differential operators $P(D)$. These extensions yields realization of the fractional powers of some pseudo-differential operators in the spirit of Caffarelli and Silvestre \cite{CSilv}. In particular every fractional power of a local pseudo-differential operator can be realized locally. We also give some properties of the relativistic Schrödinger operator $\Dsm$.

math.AP↗
arXiv2011

On the Hardy-Poincaré inequality with boundary singularities

Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.

math.AP↗
arXiv2010

A note on Hardy's inequalities with boundary singularities

Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequalities with weight function singular at the boundary of $Ø$. In particular we give sufficient conditions so that the best constant is achieved.

math.AP↗
arXiv2010

Hardy-Poincare' inequalities with boundary singularities

Let $Ø$ be a bounded domain in $\R^N$ with $0\in\deØ$ and $N\ge 2$. In this paper we study the Hardy-Poincaré inequality for maps in $H^1_0(Ω)$. In particular we give sufficient and some necessary conditions so that the best constant is achieved.

math.AP↗