Advances in Differential Equations

Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces

J. A. Griepentrog

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This text is devoted to maximal regularity results for second-order parabolic systems on Lipschitz domains of space dimension $n \ge 3$ with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial-value problems generates isomorphisms between two scales of Sobolev--Morrey spaces for solutions and right-hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower-order coefficients.

Article information

Adv. Differential Equations Volume 12, Number 9 (2007), 1031-1078.

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35D10 35R05: Partial differential equations with discontinuous coefficients or data


Griepentrog, J. A. Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces. Adv. Differential Equations 12 (2007), no. 9, 1031--1078.

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