On the lack of interior regularity of the p‐Poisson problem with p>2. Issue 6 (3rd May 2021)

• Record Type: Journal Article
• Authors: Weimar, Markus
• Is Part Of: Mathematische Nachrichten. Volume 294:Issue 6(2021)
• Page Start: 1186
• Page End: 1205
• Publication Date: 2021-05-03
• Subjects: Besov space -- nonlinear and adaptive approximation -- p‐Poisson problem -- regularity of solutions; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201900338
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: In this note we are concerned with interior regularity properties of the p ‐Poisson problem Δ 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].