Download PDF - Unique continuation for elliptic equations and an abstract differential inequalityAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Unique continuation for elliptic equations and an abstract differential inequality

Authors 1

Affiliations

  1. Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Abstract

We consider a class of elliptic equations whose leading part is the Laplacian and for which the singularities of the coefficients of lower order terms are described by a mixed Lp-norm. We prove that the zeros of the solutions are of at most finite order in the sense of a spherical L²-mean.

Bibliography

  1. N. Aronszajn, A. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417-453.
  2. B. Barcelo, C. E. Kenig, A. Ruiz and D. Sogge, Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms, Illinois J. Math. 32 (1988), 230-245.
  3. N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, Ap weights and unique continuation, Indiana Univ. Math. J. 35 (1986), 245-268.
  4. N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math. 90 (1987), 347-366.
  5. L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer, New York, 1985.
  6. L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21-64.
  7. D. Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math. 62 (1986), 118-134.
  8. D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. 121 (1985), 463-488.
  9. J. L. Kazdan, Unique continuation in geometry, Comm. Pure Appl. Math. 91 (1988), 667-681.
  10. C. E. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, in: Lecture Notes in Math. 1384, Springer, 1989, 69-90.
  11. Y. M. Kim, MIT thesis, 1989.
  12. P. D. Lax, A stability theorem for solutions of abstract differential equations, and its application to the study of the local behaviour of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 747-766.
  13. F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 93 (1990), 127-136.
  14. A. Pliś, On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. 11 (1963), 95-100.
  15. R. T. Seeley, Singular integrals on compact manifolds, Amer. J. Math. 81 (1959), 658-690.
  16. G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1) (1965), 189-258.
  17. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
  18. R. Strichartz, Multipliers for spherical harmonic expansions, Trans. Amer. Math. Soc. 167 (1972), 115-124.
  19. T. H. Wolff, Unique continuation for |Δu| ≤ V|∇u| and related problems, Rev. Mat. Iberoamericana 6 (1990), 155-200.
  20. T. H. Wolff, A property of measures in ℝn and an application to unique continuation, preprint.
Studia Mathematica
1994/108/1
Export to PBN, Opens in new tab
Pages:
5-20
Main language of publication
English
Received
1991-11-29
Accepted
1992-11-23
Published
1994
Exact and natural sciences