Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces
- Griepentrog, Jens André
2010 Mathematics Subject Classification
- 35D10 35R05 35K20
- Second order parabolic boundary value problems, instationary drift-diffusion problems, nonsmooth coefficients, unbounded lower order coefficients, mixed boundary conditions, LIPSCHITZ domains, regular sets, HARNACK-type inequality, global HOELDER continuity, maximal regularity, SOBOLEV-MORREY spaces, smooth dependence of the solutions
This text is devoted to maximal regularity results for second order parabolic systems on LIPSCHITZ domains of space dimension greater or equal than three with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of SOBOLEV-MORREY spaces for solutions and right hand sides introduced in the first part of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are HOELDER 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.
- Adv. Differential Equations, 12 (2007) pp. 1031--1078.