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- ItemParabolic equations with dynamical boundary conditions and source terms on interfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Elst, A.F.M. ter; Meyries, Martin; Rehberg, JoachimWe consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the coefficients are imposed. It is shown that the corresponding linear operator enjoys maximal parabolic regularity in a suitable Lp-setting. The linear results suffice to treat also the corresponding nondegenerate quasilinear problems.
- ItemA unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Disser, Karoline; Meyries, Martin; Rehberg, JoachimIn 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.