Weighted Korn inequalities on John domains

Weighted generalized Korn inequalities on John domains

“…Remark 3.8. Notice that the counterexample described above is not sufficient to conclude that the validity of the weighted generalized Korn inequality (5) implies that the domain is John. The equivalence between the Korn inequality (in the second case) and John's condition is studied in Jiang and Kauranen 8 and Jiang and Jiang 21 ; however, the equivalence with the generalized Korn inequality considered in the main result has not been proved yet.…”

“…The following lemma, shown in the companion article, describes the geometry of John domains in terms of Whitney decompositions and trees. Buckley et al and Vasil'eva were fundamental in the proof of this lemma.…”

“…From now on, is a bounded John domain, {Ω t } t ∈ Γ is the collection of extended Whitney cubes defined in Section 2, and Γ has a tree structure with the geometric property introduced in Lemma . The following lemma is proved in López‐García …”

“…Examples of John domains are convex domains, Lipschitz domains, and the Kock snowflake, a domain with fractal boundary. We refer the reader to previous studies [3][4][5] where some Korn-type inequalities are studied in this class of domains. The geometric John condition was introduced by Fritz John 6 and named after him by Martio and Sarvas.…”

On the weighted fractional Poincar\'e-type inequalities

“…The following fundamental result was proved in [L2,Theorem 4.4]; it shows the existence of a C-decomposition of functions subordinate to a tree covering of the domain.…”