Elliptic problems with rough boundary data in generalized Sobolev spaces

Anna Anop; Robert Denk; Aleksandr Murach

doi:10.3934/cpaa.2020286

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2021, Volume 20, Issue 2: 697-735. Doi: 10.3934/cpaa.2020286

This issue Previous Article Dual spaces of mixed-norm martingale Hardy spaces Next Article Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity

Elliptic problems with rough boundary data in generalized Sobolev spaces

1.
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01024, Ukraine
2.
University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany

^* Corresponding author
^* Corresponding author

Received: May 2020

Revised: September 2020

Early access: December 2020

Published: February 2021

The publication contains the results of studies conducted by the joint grant F81 of the National Research Fund of Ukraine and the German Research Society (DFG); competitive project F81/41686. This work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology). The first author was supported by President of Ukraine's grant for competitive project F82/45932

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Abstract

We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.

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Mathematics Subject Classification: Primary:35J40;Secondary:35R60, 46E35, 60H40.

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