Maximum principle, mean value operators and quasi boundedness in non-euclidean settings


Autoria(s): Tommasoli, Andrea
Contribuinte(s)

Lanconelli, Ermanno

Data(s)

30/06/2008

Resumo

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Formato

application/pdf

Identificador

http://amsdottorato.unibo.it/949/1/Tesi_Tommasoli_Andrea.pdf

urn:nbn:it:unibo-918

Tommasoli, Andrea (2008) Maximum principle, mean value operators and quasi boundedness in non-euclidean settings, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Matematica <http://amsdottorato.unibo.it/view/dottorati/DOT269/>, 20 Ciclo. DOI 10.6092/unibo/amsdottorato/949.

Idioma(s)

en

Publicador

Alma Mater Studiorum - Università di Bologna

Relação

http://amsdottorato.unibo.it/949/

Direitos

info:eu-repo/semantics/openAccess

Palavras-Chave #MAT/05 Analisi matematica
Tipo

Tesi di dottorato

NonPeerReviewed