Calc. Var. (2017) 56:109
Calculus of Variations
Korn’s inequality and John domains
· Aapo Kauranen
Received: 4 January 2017 / Accepted: 15 June 2017
© Springer-Verlag GmbH Germany 2017
Abstract Let ⊂ R
, n ≥ 2, be a bounded domain satisfying the separation property. We
show that the following conditions are equivalent:
(i) is a John domain;
(ii) for a ﬁxed p ∈ (1,∞), the Korn inequality holds for each u ∈ W
dx = 0, 1 ≤ i, j ≤ n,
(ii’) for all p ∈ (1,∞), (K
) holds on ;
(iii) for a ﬁxed p ∈ (1,∞), for each f ∈ L
() with vanishing mean value on ,there
exists a solution v ∈ W
) to the equation div v = f with
≤ C (, p) f
(iii’) for all p ∈ (1,∞), (DE
) holds on .
For domains satisfying the separation property, in particular, for ﬁnitely connected domains
in the plane, our result provides a geometric characterization of the Korn inequality, and
gives positive answers to a question raised by Costabel and Dauge (Arch Ration Mech Anal
217(3):873–898, 2015) and a question raised by Russ (Vietnam J Math 41:369–381, 2013).
For the plane, our result is best possible in the sense that, there exist inﬁnitely connected
domains which are not John but support Korn’s inequality.
Communicated by L. Ambrosio.
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014