Singularities in L^p-quasidisks

We study planar domains with exemplary boundary singularities of the form of cusps. A natural question is how much elastic energy is needed to flatten these cusps; that is, to remove singularities. We give, in a connection of quasidisks, a sharp integrability condition for the distortion function to answer this question. peerReviewed

Bi-Lipschitz invariance of planar BV- and W1,1-extension domains

We prove that a bi-Lipschitz image of a planar $BV$-extension domain is also a $BV$-extension domain, and that a bi-Lipschitz image of a planar $W^{1,1}$-extension domain is again a $W^{1,1}$-extension domain.

Pointwise inequalities for Sobolev functions on generalized cuspidal domains

We establish point wise inequalities for Sobolev functions on a wider class of outward cuspidal domains. It is a generalization of an earlier result by the author and his collaborators

Product of extension domains is still an extension domain

We prove the product of the Sobolev-extension domains is still a Sobolev-extension domain.

Pointwise Inequalities for Sobolev Functions on Outward Cuspidal Domains

Abstract We show that the 1st-order Sobolev spaces $W^{1,p}(\Omega _\psi ),$$1<p\leq \infty ,$ on cuspidal symmetric domains $\Omega _\psi $ can be characterized via pointwise inequalities. In particular, they coincide with the Hajłasz–Sobolev spaces $M^{1,p}(\Omega _\psi )$.

Sobolev Extension on Lp-quasidisks

AbstractIn this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.

Homeomorphisms of finite distortion : from the unit ball to cusp domains in R^{3}