Mouhamed Moustapha Fall

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.

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

We study a class of fractional elliptic problems of the form (−Δ)su = f(u) in the half-space ℝ+N := {x ∈ ℝN:x 1 > 0} with the complementary Dirichlet condition u ≡ 0 in ℝN∖ℝ +N. Under mild assumptions on the nonlinearity f, we show that bounded positive solutions are increasing in x1. For the special case f(u) = uq, we deduce nonexistence of positive bounded solutions in the case where q > 1 and q < N−1+2s N−1−2s if N ≥ 1 + 2s. We do not require integrability assumptions on the solutions we study.

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

Abstract We are concerned with hypersurfaces of ℝ N {\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 ℝ N {\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 ℝ 2 {\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.

EMBEDDED DISC-TYPE SURFACES WITH LARGE CONSTANT MEAN CURVATURE AND FREE BOUNDARIES

Given Ω ⊂ ℝ3 an open bounded set with smooth boundary ∂Ω and H ∈ ℝ, we prove the existence of embedded H-surfaces supported by ∂Ω, that is regular surfaces in ℝ3 with constant mean curvature H at every point, contained in Ω and with boundary intersecting ∂Ω orthogonally. More precisely, we prove that if Q ∈ ∂Ω is a stable stationary point for the mean curvature of ∂Ω, then there exists a family of embedded -surfaces near Q, e > 0 small, which, once dilated by a factor and suitably translated, converges to a hemisphere of radius 1 as e → 0.

Sharp Nonexistence Results for a Linear Elliptic Inequality Involving Hardy and Leray Potentials

We deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.

Liouville Theorems for a General Class of Nonlocal Operators

In this paper, we study the equation ℒu=0

Serrin's over-determined problem on Riemannian manifolds

\mathcal {L} u=0

Serrin’s overdetermined problem on the sphere

in ℝN

Hardy-Sobolev inequality with singularity a curve

\mathbb {R}^{N}

On the rate of change of the sharp constant in the Sobolev–Poincaré inequality

, where ℒ