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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

On the solvability of the Neumann problem in domains with peak


Authors: V. G. Maz'ya and S. V. Poborchiĭ
Translated by: S. V. Poborchiĭ
Original publication: Algebra i Analiz, tom 20 (2008), nomer 5.
Journal: St. Petersburg Math. J. 20 (2009), 757-790
MSC (2000): Primary 35J25
Published electronically: July 21, 2009
MathSciNet review: 2492362
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Abstract: The Neumann problem is considered for a quasilinear elliptic equation of second order in a multidimensional domain with the vertex of an isolated peak on the boundary. Under certain assumptions, the study of the solvability of this problem is reduced to a description of the dual to the Sobolev space $ W_p^1(\Omega)$ or (in the case of a homogeneous equation with nonhomogeneous boundary condition) to the boundary trace space $ TW_p^1(\Omega)$. This description involves Sobolev classes with negative smoothness exponent on Lipschitz domains or Lipschitz surfaces, and also some weighted classes of functions on the interval (0,1) of the real line. The main results are proved on the basis of the known explicit description of the spaces $ TW^1_p(\Omega)$ on a domain with an outward or inward cusp on the boundary.


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Additional Information

V. G. Maz'ya
Affiliation: Department of Mathematics, Linköping University, 58183 Linköping, Sweden
Email: vlmaz@mai.liu.se

S. V. Poborchiĭ
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospekt 28, Petrodvorets, 198504 St. Petersburg, Russia
Email: Sergei.Poborchi@paloma.spbu.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01072-3
PII: S 1061-0022(09)01072-3
Keywords: Neumann problem, Sobolev spaces, domains with cusps, boundary traces, dual spaces.
Received by editor(s): January 14, 2008
Published electronically: July 21, 2009
Additional Notes: The research of the second author was supported by RFBR (grant no. 08-01-00676-a)
Dedicated: Dedicated to V. P. Havin on the occasion of his 75th birthday
Article copyright: © Copyright 2009 American Mathematical Society