1,684 research outputs found
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 in and
on . Serrin's celebrated symmetry theorem states that,
if the normal derivative is constant on , then
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 of the torsion problem prove the estimate for some
constant depending on , where and are the radii of an
annulus containing and is a surface parallel to
at distance and sufficiently close to ;
secondly, if in addition is constant on , 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 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
Local control of Hamiltonian chaos
We review a method of control for Hamiltonian systems which is able to create
smooth invariant tori. This method of control is based on an apt modification
of the perturbation which is small and localized in phase space
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