On the maximal L p -regularity of parabolic mixed-order systems

Abstract

We study maximal L p -regularity for a class of pseudodifferential mixed-order systems on a space–time cylinder \({\mathbb{R}^n \times \mathbb{R}}\) or \({X \times \mathbb{R}}\) , where X is a closed smooth manifold. To this end, we construct a calculus of Volterra pseudodifferential operators and characterize the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms between suitable L p -Sobolev spaces of Bessel potential or Besov type. If the cross section of the space–time cylinder is compact, the inverse of a parabolic system belongs to the calculus again. As applications, we discuss time-dependent Douglis–Nirenberg systems and a linear system arising in the study of the Stefan problem with Gibbs–Thomson correction.