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We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its C 2-distance from a single sphere. The corresponding… (More)
We show that the oscillation of the scalar mean curvature of the boundary of a set controls in a quantitative way its distance from a finite family of disjoint tangent balls of equal radii. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capillarity problems.
In this work we study the problem of wave propagation in a 3-D optical fiber *. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a… (More)
We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld-Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful… (More)
In this talk we present a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition at infinity (). The most used methods to create transparent boundary conditions… (More)