Maximal regularity for a free boundary problem

This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darey's law. We prove existence of a unique maximal classical solutipn, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration. 1 Introduction and main result In this paper we investigate a class of free boundary problems, which can be described as follows. Let Given f C ~, define: = {f E BC2(R) " inf f(x)> 0}