Counterexamples to maximal regularity for operators in divergence form

Journal article (2024)

Authors

S. Bechtel Analysis - EEMCS
Connor Mooney University of California
M.C. Veraar Analysis - EEMCS
Research Group
Analysis (EEMCS) (TU Delft)
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Published Date

2024

Language

English

Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Analysis

Abstract

In this paper, we present counterexamples to maximal Lp-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 L2-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 Lp-regularity on H-1(Rd) or L2-regularity on L2(Rd).