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On the lack of interior regularity of the p ‐Poisson problem with p > 2
Author(s) -
Weimar Markus
Publication year - 2021
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201900338
Subject(s) - mathematics , smoothness , differentiable function , cover (algebra) , poisson distribution , pure mathematics , nonlinear system , besov space , mathematical analysis , statistics , functional analysis , interpolation space , physics , chemistry , quantum mechanics , engineering , gene , mechanical engineering , biochemistry
In this note we are concerned with interior regularity properties of the p ‐Poisson problemΔ p ( u ) = f with p > 2 . For all 0 < λ ≤ 1 we constuct right‐hand sides f of differentiability − 1 + λ such that the (Besov‐) smoothness of corresponding solutions u is essentially limited to 1 + λ / ( p − 1 ) . The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savaré [J. Funct. Anal. 152 (1998), 176–201], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130 (2016), 298–329].