Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 


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
Published electronically: March 21, 2013
MathSciNet review: 3014132
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • 1. Olga A. Ladyzhenskaya and Nina N. Ural′tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627
  • 2. Michael Renardy and Robert C. Rogers, An introduction to partial differential equations, Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 1993. MR 1211418
  • 3. Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
  • 4. Emilio Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in 𝑛 variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284–305 (Italian). MR 0102739
  • 5. N. Aronszajn, Boundary values of functions with finite Dirichlet integral, Conf. Partial Diff. Eq. Studies in Eigenvalue Problems, Univ. of Kansas, 1955.
  • 6. L. N. Slobodeckiĭ and V. M. Babič, On boundedness of the Dirichlet integrals, Dokl. Akad. Nauk SSSR (N.S.) 106 (1956), 604–606 (Russian). MR 0076886
  • 7. V. Maz’ya and S. Poborchi, Existence and uniqueness of an energy solution to the Dirichlet problem for the Laplace equation in the exterior of a multi-dimensional paraboloid, J. Math. Sci. (N. Y.) 172 (2011), no. 4, 532–554. Problems in mathematical analysis. No. 53. MR 2839888, 10.1007/s10958-011-0205-1
  • 8. S. Poborchi, Existence and uniqueness of a solution to the Dirichlet problem for a quasilinear equation inside and outside a paraboloid, J. Math. Sci. (N. Y.) 175 (2011), no. 3, 363–374. Problems in mathematical analysis. No. 56. MR 2839045, 10.1007/s10958-011-0351-5
  • 9. V. G. Maz’ya and S. V. Poborchii, Unique solvability of the integral equation for harmonic simple layer potential on the boundary of a domain with a peak, Vestnik St. Petersburg Univ. Math. 42 (2009), no. 2, 120–129. MR 2531449, 10.3103/S1063454109020083
  • 10. V. G. Maz′ya, Prostranstva S. L. Soboleva, Leningrad. Univ., Leningrad, 1985 (Russian). MR 807364
    Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985
  • 11. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • 12. V. G. Maz'ya and S. V. Poborchiĭ, Theorems for embedding and continuation for functions in non-Lipschitzian domains, S.-Peterburg. Gos. Univ., St. Petersburg, 2006. (Russian)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 46E35, 35G20

Retrieve articles in all journals with MSC (2010): 46E35, 35G20

Additional Information

S. V. Poborchiĭ
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia

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