Characterization of traces of smooth functions on Ahlfors regular sets

Ihnatsyeva, Lizaveta; Vähäkangas, Antti V.

doi:10.1016/j.jfa.2013.07.006

Cited by 24 publications

(27 citation statements)

References 6 publications

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“…For the extensions of minimal regularity with k = 1, and thus with α < 1 (see in addition the definition of the Besov space Def. 3.2 in [18] with the help of the normalized local best approximation in the class of polynomials P k−1 of the degree equal to k − 1) Markov's inequality is trivially satisfied for k = 1 on all closed sets of R n , and hence we do not need to impose it [25, p. 198].…”

Section: Framework Of D-sets and Markov's Local Inequalitymentioning

confidence: 99%

Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

Rozanova-Pierrat

2020

Preprint

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We present a survey of recent results of the functional analysis allowing to solve PDEs in a large class of domains with irregular boundaries. We extend the previously introduced concept of admissible domains with a d-set boundary on the domains with the boundaries on which the measure is not necessarily Ahlfors regular d-measure. This gives a generalization of Rellich-Kondrachov theorem and the compactness of the trace operator, allowing to obtain, as for a regular classical case the unicity/existence of weak solutions of Poisson boundary valued problem with the Robin boundary condition and to obtain the usual properties of the associated spectral problem.

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Section: Framework Of D-sets and Markov's Local Inequalitymentioning

confidence: 99%

Generalization of Rellich-Kondrachov theorem and trace compacteness in the framework of irregular and fractal boundaries

Rozanova-Pierrat

2020

Preprint

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“…For the extensions of minimal regularity k = 1 (see in addition the Definition of Besov space Def. 3.2 in [25] with the help of the normalized local best approximation in the class of polynomials P k−1 of the degree equal to k − 1 ) Markov's inequality is trivially satisfied.…”

Section: Introductionmentioning

confidence: 97%

Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by <i>d</i>-sets

Arfi

Rozanova-Pierrat

2019

Discrete &Amp; Continuous Dynamical Systems - S

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In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in R n , we generalize the definition of the Poincaré-Steklov operator to d -set boundaries, n − 2 < d < n , and give its spectral properties to compare to the spectra of the interior domain and also of a truncated domain, considered as an approximation of the exterior case. The well-posedness of the Robin boundary value problems for the truncated and exterior domains is given in the general framework of n -sets. The results are obtained thanks to a generalization of the continuity and compactness properties of the trace and extension operators in Sobolev, Lebesgue and Besov spaces, in particular, by a generalization of the classical Rellich-Kondrachov Theorem of compact embeddings for n and d -sets.

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“…For the case of manifolds see [12,13] and for the setting of metric spaces see [2,4,15]. Classical trace results on the Euclidean spaces can be found in [1,6,9,14,16,22,25,29,30] and studies of parabolicity on infinite networks in [26,32]. For trace results in the metric setting see [3,[18][19][20][21]33].…”

Section: Introductionmentioning

confidence: 99%