A radiation condition for the 2-D Helmholtz equation in stratified media

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 asymptotic estimates, we prove the existence of a bounded solution satisfying our radiation condition.Comment: 15 pages, 1 figur

A note on Serrin's overdetermined problem

We consider the solution of the torsion problem $-\Delta u=1$ in $\Omega$ and $u=0$ on $\partial \Omega$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_\nu$ is constant on $\partial \Omega$, then $\Omega$ must be a ball. In a recent paper, it has been conjectured that Serrin's theorem may be obtained {\it by stability} in the following way: first, for the solution $u$ of the torsion problem prove the estimate $$r_e-r_i\leq C_t\,\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partial\Omega$ and $\Gamma_t$ is a surface parallel to $\partial\Omega$ at distance $t$ and sufficiently close to $\partial\Omega$; secondly, if in addition $u_\nu$ is constant on $\partial\Omega$, show that \max_{\Gamma_t} u-\min_{\Gamma_t} u=o(C_t)\ \mbox{as} \ t\to 0^+. In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains $\Omega$ are ellipses

A note on an overdetermined problem for the capacitary potential

We consider an overdetermined problem arising in potential theory for the capacitary potential and we prove a radial symmetry result.Comment: 7 pages. This paper has been written for possible publication in a special volume dedicated to the conference "Geometric Properties for Parabolic and Elliptic PDE's. 4th Italian-Japanese Workshop", organized in Palinuro in May 201

Analytical results for 2-D non-rectilinear waveguides based on the Green's function

We consider the problem of wave propagation for a 2-D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small imperfections. Our results are based on the knowledge of a Green's function for the rectilinear case.Comment: 18 pages, 8 figure

On the shape of compact hypersurfaces with almost constant mean curvature

The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capillarity problems.Comment: 36 pages, 2 figures. In this version we have added an appendix about almost umbilical surface

Wave Propagation in a 3-D Optical Waveguide

In this paper 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 nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa.\cite{MS} The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavior of the solutions of the eigenvalue problem. Moreover, the derivation of the transform formulas needed to solve the wave propagation problem involves nontrivial calculations. The paper provides a complete description on how to construct the solution to the wave propagation problem in a 3-D optical waveguide with cylindrical symmetry. A follow-up article will study the particular cases of a step-index fiber and of a coaxial waveguide. In those cases we will obtain concrete formulas for the field and numerical examples.Comment: 35 pages, 3 figure

Wulff shape characterizations in overdetermined anisotropic elliptic problems

We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems

Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration

Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls $B_{r_e}$ and $B_{r_i}$, with the difference $r_e-r_i$ (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed in a paper by Aftalion, Busca and Reichel