Maximal Regularity, Pseudodifferential Boundary Value Problems, and the Motion of a Free Surface

We start with discussing a free boundary value problem, which describes the motion of a infinite ocean of water bounded by a fixed bottom below and a free surface above. Using the contraction mapping principle the existence of (strong) solutions locally in time can be proved once the corresponding linearized system can be solved in suitable $L^p$-Sobolev spaces in certain class of unbounded domains.

In the main part of the talk we present a method to solve this linearized system - the generalized instationary Stokes system - by means of semi-group theory. This leads to the question whether the associated Stokes operator has so-called maximal $L^p$-regularity. By a famous result due Dore and Venni it is sufficient to prove the existence of bounded imaginary powers of the Stokes operator, which can be done by pseudodifferential operator techniques.