Weighted Korn inequalities on John domains
Volume 241 / 2018
Studia Mathematica 241 (2018), 17-39
MSC: Primary 26D10; Secondary 46E35, 74B05.
DOI: 10.4064/sm8488-4-2017
Published online: 26 October 2017
Abstract
We show a weighted version of the Korn inequality on bounded Euclidean John domains, where the weights are nonnegative powers of the distance to the boundary. In this theorem, we also provide an estimate of the constant involved in the inequality which depends on the exponent that appears in the weight and a geometric condition that characterizes John domains. The proof uses a local-to-global argument based on a certain decomposition of functions.
In addition, we prove the solvability in weighted Sobolev spaces of $\mathop {\rm div}\nolimits {\bf u}=f$ on the same class of domains. In this case, the weights are nonpositive powers of the distance to the boundary. The constant appearing in this problem is also estimated.
