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Cora Sadosky: Her Mathematics, Mentorship, and Professional Contributions Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey

Authors : Ariel Barton, Svitlana Mayboroda

Published in: Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Publisher: Springer International Publishing

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Abstract

Recent years have brought significant advances in the theory of higher-order elliptic equations in non-smooth domains. Sharp pointwise estimates on derivatives of polyharmonic functions in arbitrary domains were established, followed by the higher-order Wiener test. Certain boundary value problems for higher-order operators with variable non-smooth coefficients were addressed, both in divergence form and in composition form, the latter being adapted to the context of Lipschitz domains. These developments brought new estimates on the fundamental solutions and the Green function, allowing for the lack of smoothness of the boundary or of the coefficients of the equation. Building on our earlier account of history of the subject (published in Concrete operators, spectral theory, operators in harmonic analysis and approximation). Operator Theory: Advances and Applications, vol. 236, Birkhäuser/Springer, Basel, 2014, pp. 53–93), this survey presents the current state of the art, emphasizing the most recent results and emerging open problems.

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Metadata
Title
Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey
Authors
Ariel Barton
Svitlana Mayboroda
Copyright Year
2016
Book
Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Print ISBN: 978-3-319-30959-0

Electronic ISBN: 978-3-319-30961-3

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-30961-3_4

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