Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary

Vladimir Kozlov; Vladimir Maz'ya

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 3, page 551-600
  • ISSN: 0391-173X

Abstract

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We derive an asymptotic formula of a new type for variational solutions of the Dirichlet problem for elliptic equations of arbitrary order. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.

How to cite

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Kozlov, Vladimir, and Maz'ya, Vladimir. "Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.3 (2003): 551-600. <http://eudml.org/doc/84512>.

@article{Kozlov2003,
abstract = {We derive an asymptotic formula of a new type for variational solutions of the Dirichlet problem for elliptic equations of arbitrary order. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.},
author = {Kozlov, Vladimir, Maz'ya, Vladimir},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {551-600},
publisher = {Scuola normale superiore},
title = {Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary},
url = {http://eudml.org/doc/84512},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Kozlov, Vladimir
AU - Maz'ya, Vladimir
TI - Asymptotic formula for solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients near the boundary
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 3
SP - 551
EP - 600
AB - We derive an asymptotic formula of a new type for variational solutions of the Dirichlet problem for elliptic equations of arbitrary order. The only a priori assumption on the coefficients of the principal part of the equation is the smallness of the local oscillation near the point.
LA - eng
UR - http://eudml.org/doc/84512
ER -

References

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  1. [ADN] S. Agmon – A. Douglis – L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623-727. Zbl0093.10401MR125307
  2. [Dahl] B. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986) 1119-1138. Zbl0644.35032MR859772
  3. [FJK] E. Fabes – D. Jerison – C. Kenig, Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math. 119 (1984), 121-141. Zbl0551.35024MR736563
  4. [GT] D. Gilbarg – N. Trudinger, “Elliptic Partial Differential Equations of Second Order”, (2nd ed.), Springer, 1983. Zbl0562.35001MR737190
  5. [H] L. Hörmander, “The Analysis of Linear Partial Differential Operators”, Vol. 1, Springer, 1983. Zbl0521.35002
  6. [Ken] C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, In: “Regional Conference Series in Mathematics", AMS, Providence, RI, 1994. Zbl0812.35001MR1282720
  7. [KM1] V. Kozlov – V. Maz’ya, “Differential Equations with Operator Coefficients”, (with Applications to Boundary Value Problems for Partial Differential Equations), Monographs in Mathematics, Springer-Verlag, 1999. Zbl0920.35003MR1729870
  8. [KM2] V. Kozlov – V. Maz’ya, Boundary singularities of solutions to quasilinear elliptic equations, In: “Journées Équations aux dérivées partielles", Saint-Jean-de-Monts, 31 mai-4 juin, 1999, VII-1-VII-9. Zbl1103.35324MR1718974
  9. [KM3] V. Kozlov – V. Maz’ya, Boundary behavior of solutions to linear and nonlinear elliptic equations in plane convex domains, Mathematical Research Letters 8 (2001), 1-5. Zbl0987.35050MR1825269
  10. [KMR] V. Kozlov – V. Maz’ya – J. Rossmann, “Conical singularities of solutions to elliptic equations”, Mathematical Surveys and Monographs 85 AMS, Providence, RI, 2001. Zbl0965.35003MR1788991

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