tailieunhanh - Báo cáo hóa học: "Research Article Harnack Inequality for the Schrödinger Problem Relative to Strongly Local Riemannian p-Homogeneous Forms with a Potential in the Kato Class"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Harnack Inequality for the Schrödinger Problem Relative to Strongly Local Riemannian p-Homogeneous Forms with a Potential in the Kato Class | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 24806 19 pages doi 2007 24806 Research Article Harnack Inequality for the Schrodinger Problem Relative to Strongly Local Riemannian p-Homogeneous Forms with a Potential in the Kato Class Marco Biroli and Silvana Marchi Received 17 May 2006 Revised 14 September 2006 Accepted 21 September 2006 Recommended by Ugo Gianazza We define a notion of Kato class of measures relative to a Riemannian strongly local p-homogeneous Dirichlet form and we prove a Harnack inequality on balls that are small enough for the positive solutions to a Schrodinger-type problem relative to the form with a potential in the Kato class. Copyright 2007 M. Biroli and S. 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. 1. Introduction In this paper we are interested in a Harnack inequality for the Schrodinger problem relative to strongly local Riemannian p-homogeneous forms with a potential in the Kato class. The first result in the case of Laplacian has been given by Aizenman and Simon 1 . They proved a Harnack inequality for the corresponding Schrodinger problem with a potential in the Kato measures by probabilistic methods. In 1986 Chiarenza et al. 2 gave an analitical proof of the result in the case of elliptic operators with bounded measurable coefficients. Citti et al. 3 investigated the case of the subelliptic Laplacian and in 1999 Biroli and Mosco 4 5 extended the result to the case of Riemannian strongly local Dirichlet forms we recall also that in 6 a Harnack inequality for positive harmonic functions relative to a bilinear strongly local Dirichlet form is proved . Biroli 7 considered the case p 1 for the subelliptic p-Laplacian and defining a suitable Kato class for this case he obtained again Harnack and Holder inequalities by methods