Counterexamples to the well-posedness of 𝐿^{𝑝} transmission boundary value problems for the Laplacian

Abstract

Abstract. In this note we show that the well-posedness range p ∈ (1, 2] for L p transmission boundary value problems for the Laplacian in the class of Lipschitz domains established by Escauriaza and Mitrea (2004) is sharp. Our approach relies on Mellin transform techniques for singular integrals naturally associated with the transmission problems and on a careful analysis of the L p spectra of such singular integrals.

‐solution of the Neumann, Robin and transmission problem for the scalar Oseen equation

“…It means that we look for a solution such that the nontangential maximal function of and ∇ are in ( Ω) and boundary conditions are fulfilled in the sense of nontangential limits. The goal of the paper is to prove results similar to the well-known results corresponding to the Laplace equation ( [4,5,[8][9][10][11][18][19][20][21]23,30,31,33,37,38,40,41,46,48]). We find necessary and sufficient conditions for the existence of an -solution of the Neumann and Robin problem for bounded and unbounded domains with compact Lipschitz boundary.…”

Transmission Problems for Parabolic Operators on Polygonal Domains and Applications to the Finite Element Method

“…However, in many applications, the interface is itself singular and crosses the boundary. In the elliptic case the problem has been studied using various techniques, including layer potentials for Lipschitz domains and interfaces (we refer to [24][25][26]29,42] among several works on the subject), and functional analysis techniques for corner domains [6,8,21,22,39,40,[44][45][46]50].…”

Integral equations method and the transmission problem for the stokes system

“…This problem occurs in the case of contact of two media with different material constants. Very fruitful is to study this problem using the integral equation method (see [4], [35], [12], [11], [13], [34], [5]). Many papers study the Brinkman transmission problem and the Stokes-Brinkman transmission problem by the integral equation method ( [21], [22], [23], [25], [20], [24]).…”