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Sib. Èlektron. Mat. Izv., 2013, Volume 10, Pages 335–377 (Mi semr417)  

This article is cited in 2 scientific papers (total in 2 papers)

Differentical equations, dynamical systems and optimal control

Error bound for a generalized M. A. Lavrentiev's formula via the norm in a fractional Sobolev space

A. I. Parfenov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: We generalize M. A. Lavrentiev's approximate formula for the conformal mapping of the perturbed half-plane onto the half-plane. The generalization concerns harmonic functions and their derivatives in locally perturbed half-spaces (Lipschitz epigraphs). For both formulas, we obtain remainder estimates involving the square of the norm of the perturbing function in the fractional homogeneous Sobolev space $\dot{H}^{1/2}$. By the Kashin–Besov–Kolyada inequality, these estimates imply pointwise stability bounds in terms of the Lebesgue measure. Moreover, we prove the joint analyticity of the above-named harmonic functions with respect to the perturbing parameter and the space variables and justify a result on the interpolation between $L^1$ and homogeneous Slobodetskii spaces which is essentially due to A. Cohen.

Keywords: harmonic function, Lavrentiev formula, perturbed domain, quantitative stability, remainder estimate.

Full text: PDF file (851 kB)
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UDC: 517.95
MSC: 35C20
Received October 25, 2012, published April 14, 2013

Citation: A. I. Parfenov, “Error bound for a generalized M. A. Lavrentiev's formula via the norm in a fractional Sobolev space”, Sib. Èlektron. Mat. Izv., 10 (2013), 335–377

Citation in format AMSBIB
\Bibitem{Par13}
\by A.~I.~Parfenov
\paper Error bound for a generalized M.\,A.~Lavrentiev's formula via the norm in a fractional Sobolev space
\jour Sib. \`Elektron. Mat. Izv.
\yr 2013
\vol 10
\pages 335--377
\mathnet{http://mi.mathnet.ru/semr417}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Parfenov, “Discrete Hölder estimates for a parametrix variation”, Siberian Adv. Math., 25:3 (2015), 209–229  mathnet  crossref  mathscinet
    2. A. I. Parfenov, “Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem”, Siberian Adv. Math., 27:4 (2017), 274–304  mathnet  crossref  crossref  elib
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