×

An elementary proof of Harnack’s inequality for Schrödinger operators and related topics. (English) Zbl 0697.35017

For \(G\subset {\mathbb{R}}^ N\) an open set, the author deals with distributional solutions of \(-\Delta u+Vu=f\) where \(V,f\in K^ N_{loc}(G)\). Here, \(K^ N(G)\) denotes a class of functions introduced for Schrödinger equations by T. Kato. Main topics are the continuity of distributional solutions and the validity of Harnack’s inequality as well as stronger versions of it which were (for \(f=0)\) originally established by Aizenman and Simon using analytic and probabilistic tools. Meanwhile purely analytic approaches to those results are known.
In the present paper, a new and straightforward approach is presented which is essentially based on local representations of the solution u by means of fundamental solution and Green’s function to the Laplacian. Remembering that the class \(K^ N(G)\) is defined in terms of the fundamental solution of the Laplacian, the new approach appears very natural.
Reviewer: W.Velte

MSC:

35B45 A priori estimates in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35A08 Fundamental solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aizenman, M., Simon, B.: Brownian motion and harnack inequality for Schrödinger operators. Commun. Pure Appl. Math.35, 209–273 (1982) · Zbl 0475.60063 · doi:10.1002/cpa.3160350206
[2] Chiarenza, F., Fabes, E., Garofalo, N.: Harnack’s inequality for Schrödinger operators and the continuity of solutions. Proc. Am. Math. Soc.98, 415–425 (1986) · Zbl 0626.35022
[3] Fabes, E.B., Stroock, D.W.: TheL p -integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. Duke Math J.51, 997–1016 (1984) · Zbl 0567.35003 · doi:10.1215/S0012-7094-84-05145-7
[4] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin Heidelberg New York: Springer 1977 · Zbl 0361.35003
[5] Hellwig, G.: Partial differential equations. An Introduction, Stuttgart: Teubner 1977 · Zbl 0361.35001
[6] Hinz, A.M., Kalf, H.: Subsolution estimates and Harnack’s inequality for Schrödinger operators. (to appear) · Zbl 0779.35026
[7] Ikebe, T., Kato, T.: Uniqueness of self-adjoint extensions of singular elliptic differential operators. Arch. Ration. Mech. Anal.9, 77–92 (1962) · Zbl 0103.31801 · doi:10.1007/BF00253334
[8] Kato, T.: Schrödinger operators with singular potentials. Isr. J. Math.13, 135–148 (1972) · Zbl 0246.35025 · doi:10.1007/BF02760233
[9] Leinfelder, H., Simader, C.G.: Schrödinger operators with singular magnetic vector potentials. Math. Z.176, 1–19 (1981) · Zbl 0468.35038 · doi:10.1007/BF01258900
[10] Schechter, M.: Spectra of partial differential operators. Amsterdam, London: North-Holland 1971 · Zbl 0225.35001
[11] Wiegner, M.: Private communication 1988
[12] Wienholtz, W.: Halbbeschränkte partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus. Math. Ann.135, 50–80 (1958) · Zbl 0142.37701 · doi:10.1007/BF01350827
[13] Witte, J.: Über die Regularität der Spektralschar eines singulären elliptischen. Differentialoperators. Math Z.107, 116–126 (1968) · Zbl 0167.11101 · doi:10.1007/BF01111024
[14] Zhao, Z.: Conditional gauge with unbounded potential. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 13–18 (1983) · Zbl 0521.60074 · doi:10.1007/BF00534990
[15] Zhao, Z.: Green function for Schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl.116, 309–334 (1986) · Zbl 0608.35012 · doi:10.1016/S0022-247X(86)80001-4
[16] Zhao, Z.: Uniform boundedness of conditional gauge and Schrödinger equations. Commun. Math. Phys.93, 19–31 (1984) · Zbl 0545.35087 · doi:10.1007/BF01218637
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.