Elliptic and Parabolic Equations - Abstract

Nicaise, Serge

Singular behavior of the solution of the heat equation in weighted $L^p$-Sobolev spaces

We consider the heat equation on a polygonal domain $Omega$ of the plane in weighted $L^p$-Sobolev spaces. The classical Fourier transform techniques do not allow to handle such a general case. Hence we use the theory of sums of operators. The first step is to study the Helmholtz equation [ (1)quad -Delta u+zu=g hbox in Omega, ] with Dirichlet boundary conditions, where $z$ is a complex number such that $Re zgeq 0$. Here $g$ belongs to $L^p_mu(Omega)=v in L^p_loc(Omega): r^mu vin L^p(Omega),$ with a real parameter $mu$ and $r(x)$ the distance from $x$ to the set of corners of $Omega$. We give sufficient conditions on $mu, p$ and $Omega$ that guarantee that problem (1) has a unique solution $uin H1_0(Omega)$ that admits a decomposition into a regular part in weighted $L^p$-Sobolev spaces and an explicit singular part. We further obtain some estimates where the explicit dependence on $ z $ is given.
Common work with Colette De Coster.