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On regularization of the Cauchy problem for elliptic systems in weighted Sobolev spaces

Yulia Shefer and Alexander Shlapunov EMAIL logo

Abstract

We consider the ill-posed Cauchy problem in a bounded domain 𝒟 of n for an elliptic differential operator 𝒜(x,) with data on a relatively open subset S of the boundary 𝒟. We do it in weighted Sobolev spaces Hs,γ(𝒟) containing the elements with prescribed smoothness s and growth near S in 𝒟, controlled by a real number γ. More precisely, using proper (left) fundamental solutions of 𝒜(x,), we obtain a Green-type integral formula for functions from Hs,γ(𝒟). Then a Neumann-type series, constructed with the use of iterations of the (bounded) integral operators applied to the data, gives a solution to the Cauchy problem in Hs,γ(𝒟) whenever this solution exists.

MSC 2010: 35JXX; 35NXX

Award Identifier / Grant number: N 1.2604.2017/PCh

Funding statement: The authors were supported by Grant N 1.2604.2017/PCh of the Ministry of Education and Science of the Russian Federation.

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Received: 2018-02-07
Revised: 2019-03-03
Accepted: 2019-04-02
Published Online: 2019-06-14
Published in Print: 2019-12-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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