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Multilayer potentials
for higher-order systems in rough domains
Gustavo Hoepfner, Paulo Liboni, Dorina Mitrea, Irina Mitrea and Marius Mitrea |
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Vol. 14 (2021), No. 4, 1233–1308
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Abstract |
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We initiate the study of multilayer potential operators associated with any given homogeneous constant-coefficient higher-order elliptic system in an open set satisfying additional assumptions of a geometric measure theoretic nature. We develop a Calderón–Zygmund-type theory for this brand of singular integral operators acting on Whitney arrays, starting with the case when is merely of locally finite perimeter and then progressively strengthening the hypotheses by ultimately assuming that is a uniformly rectifiable domain (which is the optimal setting where singular integral operators of principal value type are known to be bounded on Lebesgue spaces), and conclude by indicating how this body of results is significant in the context of boundary value problems for the higher-order system in such a domain . |
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Keywords
higher-order system, multilayer operator, boundary layer
potential, Calderón–Zygmund operator, principal value
singular integral operator, set of locally finite
perimeter, Ahlfors regular set, uniformly rectifiable set,
divergence theorem, nontangential maximal operators,
nontangential boundary trace, Whitney arrays, boundary
Sobolev space, Carleson measure, Dirichlet boundary
problem, regularity boundary problem
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Mathematical Subject Classification 2010
Primary: 31B10, 31B25, 35C15, 42B20, 42B37, 49Q15
Secondary: 35A01, 35A02, 35J58, 42B25, 42B35, 42B35, 45E05,
45P05, 47B38, 47G10
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Milestones
Received: 7 May 2019
Revised: 10 November 2019
Accepted: 20 December 2019
Published: 6 July 2021
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Authors |
| Gustavo Hoepfner | |
| Departamento de Matemática Universidade Federal de São Carlos São Carlos Brazil |
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| Paulo Liboni | |
| Departamento de Matemática Universidade Estadual de Londrina Londrina Brazil |
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| Dorina Mitrea | |
| Department of Mathematics Baylor University Waco, TX United States |
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| Irina Mitrea | |
| Department of Mathematics Temple University Philadelphia, PA United States |
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| Marius Mitrea | |
| Department of Mathematics Baylor University Waco, TX United States |
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