Stratified Lie Groups and Potential Theory for Their Sub-Laplacians [electronic resource] /
Bonfiglioli, Andrea.author.
Lanconelli, Ermanno.author.
Uguzzoni, Francesco.author.
The existence, for every sub-Laplacian, of a homogeneous fundamental solution smooth out of the origin, plays a crucial role in the book. This makes it possible to develop an exhaustive Potential Theory, almost completely parallel to that of the classical Laplace operator. This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. In recent years, sub-Laplacian operators have received considerable attention due to their special role in the theory of linear second-order PDE's with semidefinite characteristic form. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra nor in differential geometry. It is thus addressed, besides PhD students, to junior and senior researchers in different areas such as: partial differential equations; geometric control theory; geometric measure theory and minimal surfaces in stratified Lie groups.
Description based upon print version of record.
Includes bibliographical references (p. [773]-787) and indexes.
Elements of Analysis of Stratified Groups -- Stratified Groups and Sub-Laplacians -- Abstract Lie Groups and Carnot Groups -- Carnot Groups of Step Two -- Examples of Carnot Groups -- The Fundamental Solution for a Sub-Laplacian and Applications -- Elements of Potential Theory for Sub-Laplacians -- Abstract Harmonic Spaces -- The ?-harmonic Space -- ?-subharmonic Functions -- Representation Theorems -- Maximum Principle on Unbounded Domains -- ?-capacity, ?-polar Sets and Applications -- ?-thinness and ?-fine Topology -- d-Hausdorff Measure and ?-capacity -- Further Topics on Carnot Groups -- Some Remarks on Free Lie Algebras -- More on the Campbell-Hausdorff Formula -- Families of Diffeomorphic Sub-Laplacians -- Lifting of Carnot Groups -- Groups of Heisenberg Type -- The Carathéodory-Chow-Rashevsky Theorem -- Taylor Formula on Homogeneous Carnot Groups.
The existence, for every sub-Laplacian, of a homogeneous fundamental solution smooth out of the origin, plays a crucial role in the book. This makes it possible to develop an exhaustive Potential Theory, almost completely parallel to that of the classical Laplace operator. This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. In recent years, sub-Laplacian operators have received considerable attention due to their special role in the theory of linear second-order PDE's with semidefinite characteristic form. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra nor in differential geometry. It is thus addressed, besides PhD students, to junior and senior researchers in different areas such as: partial differential equations; geometric control theory; geometric measure theory and minimal surfaces in stratified Lie groups.
Algebra.
Partial differential equations.
Potential theory (Mathematics).
Topological groups.
Lie groups.
Algebra.
Partial Differential Equations.
Potential Theory.
Topological Groups, Lie Groups.


Springer Monographs in Mathematics