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 $M^{1,p}(\Omega _\psi )$.
Our main result Theorem 1.1 gives the following functional property of the class of W 1,p -extension domains. Let Ω 1 ⊂ R n and Ω 2 ⊂ R m both be W 1,p -extension domains for some 1 < p ≤ ∞. We prove that Ω 1 × Ω 2 ⊂ R n+m is also a W 1,p -extension domain. We also establish the converse statement.
The concept of hyperelastic deformations of bi-conformal energy is developed as an extension of quasiconformality. These are homeomorphisms h : X onto − − → Y between domains X, Y ⊂ R n of the Sobolev class W 1,n loc (X, Y) whose inverse f def 2010 Mathematics Subject Classification. Primary 30C65; Secondary 46E35, 58C07.
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.
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