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Unique solvability of the Dirichlet problem for the equation $ \Delta_p u=0$ in the exterior of a paraboloid


Author: S. V. Poborchiĭ
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 493-512
MSC (2010): Primary 46E35; Secondary 35G20
DOI: https://doi.org/10.1090/S1061-0022-2013-01250-7
Published electronically: March 21, 2013
MathSciNet review: 3014132
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Abstract: The Dirichlet problem

$\displaystyle -\textup {div}(\vert\nabla u\vert^{p-2}\nabla u)=0 \ $$\displaystyle \mbox { in } \ \Omega , \ u\big \vert _{\partial \Omega }=f, $

is considered in the exterior of an $ n$-dimensional paraboloid, $ p\in (1,n)$. The space of the traces $ u\big \vert _\Gamma $ on the boundary of the paraboloid for functions $ u$ in the class $ L_p^1$ is described explicitly. This implies necessary and sufficient conditions for the existence and uniqueness of a solution to the Dirichlet problem.

References [Enhancements On Off] (What's this?)

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Additional Information

S. V. Poborchiĭ
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: poborchi@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01250-7
Keywords: Dirichlet problem for an unbounded domain, a domain with infinite locally-Lipschitz boundary, traces of functions with gradient in $L_p$ on a locally-Lipschitz boundary
Received by editor(s): September 13, 2011
Published electronically: March 21, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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