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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

Published: 14 February 2007

Abstract

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.