Maximal regularity of parabolic transmission problems
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Journal of Evolution Equations
Maximal regularity of parabolic transmission problems Herbert Amann
Dedicated to Matthias Hieber, a pioneer of maximal regularity, on the occasion of his sixtieth birthday Abstract. Linear reaction–diffusion equations with inhomogeneous boundary and transmission conditions are shown to possess the property of maximal L p regularity. The new feature is the fact that the transmission interface is allowed to intersect the boundary of the domain transversally.
1. Introduction The emerging and understanding of the theory of maximal regularity for parabolic differential equations, which took place within the last three or so decades, has provided a firm basis for a successful handling of many challenging nonlinear problems. Among them, phase transition issues play a particularly prominent role. The impressive progress which has been made in this field with the help of maximal regularity techniques is well documented in the book by Prüss and Simonett [32]. The reader may also consult the extensive list of references and the ‘Bibliographic Comments’ in [32] for works of other authors and historical developments. The relevant mathematical setup is usually placed in the framework of parabolic equations in bounded Euclidean domains, the interface being modeled as a hypersurface. In most works known to the author, it is assumed that the interface lies in the interior of the domain. Noteworthy exceptions are the papers by Wilke [37], Prüss et al. [33], Abels et al. [1], and Rauchecker [34] who study various important parabolic free boundary problems, presupposing that the membrane makes a ninety-degree boundary contact. In addition, in all of them, except for [1], a capillary (i.e., cylindrical) geometry is being studied. The same ninety-degree condition is employed by Garcke and Rauchecker [25] who carry out a linearized stability computation at a stationary solution of a Mullins–Sekerka flow in a two-dimensional bounded domain. Mathematics Subject Classification: 35K10, 35K57, 35K65, 58J32 Keywords: Linear reaction–diffusion equations, Inhomogeneous boundary and transmission conditions, Interfaces with boundary intersection, Maximal regularity, Riemannian manifolds with bounded geometry and singularities, Weighted Sobolev spaces.
H. Amann
J. Evol. Equ.
The assumption of the ninety-degree contact considerably simplifies the analysis since it allows to use reflection arguments. This does not apply in the case of general transversal intersection. The only paper, we are aware of, in which a general contact angle is being considered is the one by Laurençot and Walker [28]. These authors establish the unique solvability in the strong L 2 sense of a two-dimensional stationary transmission problem taking advantage of a particularly favorable geometric setting. Elliptic problems with boundary and transmission conditions have also been investigated in a series of papers by Nistor et al. [21,29–31]. The motivation for these works stems from the desire to get optimal convergence rates for approximatio
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