On the lack of interior regularity of the p-Poisson problem with p>2

07/30/2019

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by   Markus Weimar, et al.

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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. 152:176-201, 1998], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329, 2016]. Keywords: Nonlinear and adaptive approximation, Besov space, regularity of solutions, p-Poisson problem.