Harnack’s inequality for degenerate Schrödinger operators

Gutiérrez, Cristian E.

doi:10.1090/S0002-9947-1989-0948190-6

Harnack’s inequality for degenerate Schrödinger operators
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by Cristian E. Gutiérrez PDF

Trans. Amer. Math. Soc. 312 (1989), 403-419 Request permission

Abstract:

We prove a Harnack inequality for nonnegative weak solutions of certain Schrödinger equations of the form $Lu - Vu = 0$ where $L$ is a second order degenerate elliptic operator in divergence form and $V$ is a potential in certain class.

References

F. Chiarenza, E. Fabes, and N. Garofalo, Harnack’s inequality for Schrödinger operators and the continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), no. 3, 415–425. MR 857933, DOI 10.1090/S0002-9939-1986-0857933-4
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
Gianni Dal Maso and Umberto Mosco, Wiener criteria and energy decay for relaxed Dirichlet problems, Arch. Rational Mech. Anal. 95 (1986), no. 4, 345–387. MR 853783, DOI 10.1007/BF00276841
E. Fabes, D. Jerison, and C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vi, 151–182 (English, with French summary). MR 688024, DOI 10.5802/aif.883
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
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), no. 4, 997–1016. MR 771392, DOI 10.1215/S0012-7094-84-05145-7
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
Benjamin Muckenhoupt and Richard L. Wheeden, Weighted bounded mean oscillation and the Hilbert transform, Studia Math. 54 (1975/76), no. 3, 221–237. MR 399741, DOI 10.4064/sm-54-3-221-237
Martin Schechter, Spectra of partial differential operators, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. MR 869254