Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Harnack's inequality for Schrödinger operators and the continuity of solutions


Authors: F. Chiarenza, E. Fabes and N. Garofalo
Journal: Proc. Amer. Math. Soc. 98 (1986), 415-425
MSC: Primary 35B99; Secondary 35D10, 35J15
DOI: https://doi.org/10.1090/S0002-9939-1986-0857933-4
MathSciNet review: 857933
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a uniform Harnack inequality for nonnegative solutions of $ Au = Vu$, where $ A$ is a second order elliptic operator in divergence form, and $ V$ belongs to the Stummel class of potentials. As a consequence we obtain the continuity of a general weak solution. These results extend the previous work of Aizenman and Simon for $ \Delta u = Vu$.


References [Enhancements On Off] (What's this?)

  • [1] M. Aizenman and B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209-271. MR 644024 (84a:35062)
  • [2] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
  • [3] G. Dal Maso and U. Mosco, Weiner criteria and energy decay for relaxed Dirichlet problems, preprint. MR 853783 (87m:35021)
  • [4] E. De Giorgi, Sulla differentiabilità e l'analiticità degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (3) 3 (1957), 25-43. MR 0093649 (20:172)
  • [5] E. B. Fabes and D. W. Stroock, The $ {L^p}$-integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), 997-1016. MR 771392 (86g:35057)
  • [6] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 577-591. MR 0131498 (24:A1348)
  • [7] W. Littman, G Stampacchia and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 45-79. MR 0161019 (28:4228)
  • [8] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. MR 0159138 (28:2356)
  • [9] B. Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math. 49 (1974), 101-106. MR 0350297 (50:2790)
  • [10] J. Nash, Continuity of the solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931-954. MR 0100158 (20:6592)
  • [11] M. V. Safanov, Harnack's inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math. 21 (1983), 851-863.
  • [12] M. Schechter, Spectra of partial differential operators, North-Holland, 1971. MR 0447834 (56:6144)
  • [13] N. S. Trudinger, Local estimates for subsolutions and supersolutions of general second order elliptic quasilinear equations, Invent. Math. 161 (1980), 67-79. MR 587334 (81m:35058)
  • [14] Z. Zhao, Conditional guage with unbounded potential, Z. Wahrsch. Verw. Gebiete 65 (1983), 13-18. MR 717929 (86m:60188b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35B99, 35D10, 35J15

Retrieve articles in all journals with MSC: 35B99, 35D10, 35J15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0857933-4
Keywords: Harnack's inequality, Schrödinger equation
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society