On the solvability of the Neumann problem in domains with peak

Maz’ya, V.; Poborchiĭ, S.

doi:10.1090/S1061-0022-09-01072-3

On the solvability of the Neumann problem in domains with peak
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by V. G. Maz’ya and S. V. Poborchiĭ
Translated by: S. V. Poborchiĭ

St. Petersburg Math. J. 20 (2009), 757-790

DOI: https://doi.org/10.1090/S1061-0022-09-01072-3

Published electronically: July 21, 2009

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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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