Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in H\"older spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.Comment: 33 page

Transmission Robin problem for singular p(x)-Laplacian equation in a cone

We study the behavior near the boundary angular or conical point of weak solutions to the transmission Robin problem for an elliptic quasi-linear second-order equation with the variable p(x)-Laplacian

Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.Comment: 54 page

Inverse scattering for periodic structures: Stability of polygonal interfaces

We consider the two-dimensional TE and TM diffraction problems for a time harmonic plane wave incident on a periodic grating structure. An inverse diffraction problem is to determine the grating profile from measured reflected and transmitted waves away from the structure. We present a new approach to this problem which is based on the material derivative with respect to the variation of the dielectric coefficient. This leads to local stability estimates in the case of interfaces with corner points