WIAS Preprint No. 1905, (2013)

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces



Authors

  • Disser, Karoline
  • Meyries, Martin
  • Rehberg, Joachim

2010 Mathematics Subject Classification

  • 35K20 35M13 35R05 35K65 35R01

Keywords

  • Parabolic equations, mixed boundary conditions, dynamical boundary conditions, Lipschitz domain, degenerate diffusion, surface diffusion, power weights, maximal parabolic Lp-regularity

Abstract

In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.

Appeared in

  • J. Math. Anal. Appl., 430 (2015) pp. 1102--1123.

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