Maximal L _p -regularity for a linear three-phase problem of parabolic–elliptic type

Abstract

We prove maximal L p -regularity for a three-phase problem consisting of strongly coupled parabolic–elliptic equations with inhomogeneous data. This problem is related to a nonlinear problem which arises in chemically reacting systems incorporating electromigration. Particular features are a transmission condition and a jump condition on the boundary, which couple all unknowns. By means of localization the problem is reduced to model problems in full and half-space. To solve model problems, we make use of Dore–Venni Theory, real interpolation and the Mikhlin multiplier theorem in the operator-valued version. Here it is crucial to find conditions on the data that are necessary and sufficient for maximal L p -regularity of the respective solution.