Search results for "Primary 46E35"

showing 2 items of 2 documents

Density of Lipschitz functions in energy

2020

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space $N^{1,p}(X)$ holds for all $p\in [1,\infty)$ whenever the space $X$ is complete and separable and the measure is Radon and finite on balls. Emphatically, $p=1$ is allowed. We also give a few corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. It also yields a new approximation technique, which has not appeared in prior work. Notable with all of this is that we do not use any form of Poincar\'e inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of existing literature on th…

Primary 46E35 Secondary 30L99 26B30 28A12Mathematics - Classical Analysis and ODEsApplied MathematicsClassical Analysis and ODEs (math.CA)FOS: MathematicsfunktionaalianalyysiAnalysis
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Bi-Sobolev extensions

2022

We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling-Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling-Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.

Sobolev extensionskvasikonformikuvauksetMathematics - Complex VariablesPrimary 46E35 30C62. Secondary 58E20FOS: Mathematicsharmonic extensionquasiconformal mapping and mapping of finite distortionSobolev homeomorphismsComplex Variables (math.CV)Beurling-Ahlfors extension
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