Counterexamples to maximal regularity for operators in divergence form

In this paper, we present counterexamples to maximal L p -regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L 2 -regularity on H - 1 under a coerc...

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Bibliographic Details
Published in:Archiv der Mathematik 2024-08, Vol.123 (2), p.199-209
Main Authors: Bechtel, Sebastian, Mooney, Connor, Veraar, Mark
Format: Article
Language:English
Subjects:
Coercivity
Mathematics
Mathematics and Statistics
Operators
Parabolic differential equations
Partial differential equations
Regularity
Time dependence
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Summary:In this paper, we present counterexamples to maximal L p -regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal L 2 -regularity on H - 1 under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal L p -regularity on H - 1 ( R d ) or L 2 -regularity on L 2 ( R d ) .
ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-024-02014-9