The Heat Equaton with General Periodic Boundary Conditions

Abstract

In this paper we address the well posedness of the linear heat equation under general periodic boundary conditions in several settings depending on the properties of the initial data. We develop an Lq theory not based on separation of variables and use techniques based on uniform spaces. We also allow less directions of periodicity than the dimension of the problem. We obtain smoothing estimates on the solutions. Also, based on symmetry arguments, we handle Dirichlet or Neumann boundary conditions in some faces of the unit cell.