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

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.

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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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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
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
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