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The mixed problem for harmonic functions in polyhedra of \(\mathbb R^3\). (English) Zbl 1160.35022

Mitrea, Dorina (ed.) et al., Perspectives in partial differential equations, harmonic analysis and applications. A volume in honor of Vladimir G. Maz’ya’s 70th birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4424-3/hbk). Proceedings of Symposia in Pure Mathematics 79, 407-422 (2008).
The author investigated the mixed problem for harmonic functions in Polyhedral domains. The problem comes form the study on the mixed boundary value problem for harmonic functions in creased Lipschitz domains. Here the domain is called a compact polyhedral domain, which may be not a Lipschitz domain. The author gave the existence and uniqueness of the solution to the mixed problem with data in Sobolev spaces. The paper can be seen as a continuation of G. C. Verchota and R. M. Brown’s work. The most interesting work in the paper is also to give examples at last to explain the necessity of the geometric hypotheses, i.e.,the third condition of a compact polyhedral domain.
For the entire collection see [Zbl 1149.43002].

MSC:

35J30 Higher-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
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