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This article is part of the series Harnack's Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations.

Open Access Research Article

Harnack Inequality for the Schrödinger Problem Relative to Strongly Local Riemannian -Homogeneous Forms with a Potential in the Kato Class

Marco Biroli12* and Silvana Marchi3

Author Affiliations

1 Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy

2 Accademia Nazionale delle Scienze detta dei XL, Via L. Spallanzani 7, 00161 Roma, Italy

3 Dipartimento di Matematica, Università di Parma, Viale Usberti 53/A, 43100 Parma, Italy

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Boundary Value Problems 2007, 2007:024806  doi:10.1155/2007/24806


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/024806


Received:17 May 2006
Revisions received:14 September 2006
Accepted:21 September 2006
Published:14 February 2007

© 2007 Biroli and Marchi

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We define a notion of Kato class of measures relative to a Riemannian strongly local -homogeneous Dirichlet form and we prove a Harnack inequality (on balls that are small enough) for the positive solutions to a Schrödinger-type problem relative to the form with a potential in the Kato class.

References

  1. Aizenman, M, Simon, B: Brownian motion and Harnack inequality for Schrödinger operators. Communications on Pure and Applied Mathematics. 35(2), 209–273 (1982). Publisher Full Text OpenURL

  2. Chiarenza, F, Fabes, E, Garofalo, N: Harnack's inequality for Schrödinger operators and the continuity of solutions. Proceedings of the American Mathematical Society. 98(3), 415–425 (1986)

  3. Citti, G, Garofalo, N, Lanconelli, E: Harnack's inequality for sum of squares of vector fields plus a potential. American Journal of Mathematics. 115(3), 699–734 (1993). Publisher Full Text OpenURL

  4. Biroli, M: Weak Kato measures and Schrödinger problems for a Dirichlet form. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I. 24, 197–217 (2000)

  5. Biroli, M, Mosco, U: Sobolev inequalities on homogeneous spaces. Potential Analysis. 4(4), 311–324 (1995). Publisher Full Text OpenURL

  6. Biroli, M, Mosco, U: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Annali di Matematica Pura ed Applicata. Serie Quarta. 169(1), 125–181 (1995). Publisher Full Text OpenURL

  7. Biroli, M: Nonlinear Kato measures and nonlinear subelliptic Schrödinger problems. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I. 21, 235–252 (1997)

  8. Malý, J: Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points. Commentationes Mathematicae Universitatis Carolinae. 37(1), 23–42 (1996)

  9. Malý, J, Ziemer, WP: Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Surveys and Monographs,p. xiv+291. American Mathematical Society, Rhode Island (1997)

  10. Biroli, M, Marchi, S: Oscillation estimates relative to -homogeneous forms and Kato measures data. to appear in Le Matematiche

  11. Biroli, M: Strongly local nonlinear Dirichlet functionals and forms. to appear in Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni

  12. Biroli, M, Vernole, PG: Strongly local nonlinear Dirichlet functionals and forms. Advances in Mathematical Sciences and Applications. 15(2), 655–682 (2005)

  13. Fukushima, M, Ōshima, Y, Takeda, M: Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics,p. x+392. Walter de Gruyter, Berlin (1994)

  14. Coifman, RR, Weiss, G: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Mathematics,p. v+160. Springer, Berlin (1971)

  15. Malý, J, Mosco, U: Remarks on measure-valued Lagrangians on homogeneous spaces. Ricerche di Matematica. 48(suppl.), 217–231 (1999)

  16. Kato, T: Schrödinger operators with singular potentials. Israel Journal of Mathematics. 13, 135–148 (1973) (1972). Publisher Full Text OpenURL

  17. Biroli, M, Mosco, U: Kato space for Dirichlet forms. Potential Analysis. 10(4), 327–345 (1999). Publisher Full Text OpenURL

  18. Biroli, M: Schrödinger type and relaxed Dirichlet problems for the subelliptic -Laplacian. Potential Analysis. 15(1-2), 1–16 (2001)

  19. Biroli, M, Tchou, NA: Nonlinear subelliptic problems with measure data. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni. Serie V. Parte I. 23, 57–82 (1999)

  20. Biroli, M, Vernole, P: Harnack inequality for harmonic functions relative to a nonlinear -homogeneous Riemannian Dirichlet form. Nonlinear Analysis. 64(1), 51–68 (2006). Publisher Full Text OpenURL