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, 20 Ciclo. DOI 10.6092/unibo/amsdottorato/949.
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Abstract
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.
Abstract
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.
Tipologia del documento
Tesi di dottorato
Autore
Tommasoli, Andrea
Supervisore
Dottorato di ricerca
Ciclo
20
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
maximum principle x-elliptic mean value operators quasi boundedness convexity
URN:NBN
DOI
10.6092/unibo/amsdottorato/949
Data di discussione
30 Giugno 2008
URI
Altri metadati
Tipologia del documento
Tesi di dottorato
Autore
Tommasoli, Andrea
Supervisore
Dottorato di ricerca
Ciclo
20
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
maximum principle x-elliptic mean value operators quasi boundedness convexity
URN:NBN
DOI
10.6092/unibo/amsdottorato/949
Data di discussione
30 Giugno 2008
URI
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