Stratified Lie Groups and Potential Theory for Their Sub-Laplacians / Edition 1

by Andrea Bonfiglioli, Ermanno Lanconelli, Francesco Uguzzoni

Overview

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.

Product Details

ISBN-13:	9783642090998
Publisher:	Springer Berlin Heidelberg
Publication date:	11/10/2010
Series:	Springer Monographs in Mathematics
Edition description:	Softcover reprint of hardcover 1st ed. 2007
Pages:	802
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

About the Author

1) ERMANNO LANCONELLI:

—Education and Undergraduate Studies: Dec. 1966, Universita' di Bologna (Mathematics).

Career/Employment:

1975-present: Full Professor of Mathematical Analysis at Dipartimento di Matematica, Universita' di Bologna (Italy); Member of the "Accademia dell'Istituto di Bologna" and of the "Accademia delle Scienze, Lettere ed Arti di Modena".

1968-1975: Theaching Assistant at Istituto di Matematica, Universita' di Bologna.

—Academic activity:

Director of the Istituto di Matematica di Bologna(1978/80),

Director of the Undergraduate Mathematics Program, University of Bologna (1990/1999, 2000-2002, 2006-present)

Director of PHD program, University of Bologna (1986/91, 1997/2000)

—INVITATIONS:

-University of Minnesota, Minneapolis (USA)

-University of Purdue, West La Fayette, Indiana (USA)

-Temple University, Philadelphia, Pennsylvania (USA)

-Rutgers University, New Brunswick, New Jersey (USA)

-University of Bern, Switzerland

— Specialization main fields: Partial Differential Equations, Potential

Theory

—CURRENT RESEARCH INTEREST:

Second order linear and nonlinear partial differential equations with non- negative characteristic form and application to complex geometry and diffusion processes.

Potential Theory and Harmonic Analysis in sub-riemannian settings.

Real analysis and geometric methods.

—EDITORIAL BOARD: Nonlinear Differential Equations and Applications, Birkhauser.

—PUBLICATIONS: More than 70 papers in refereed journals.

2) UGUZZONI FRANCESCO:

—Education and Undergraduate Studies: Dec. 1994, Universita' di Bologna (Mathematics)

Career/Employment:

February 2000: Ph.D. in Mathematical Analysis at Dipartimento di Matematica, Universita' di Bologna (Italy).

October 1998: Assistant Professor at Dipartimento di Matematica, Universita' di Bologna.

—CURRENT RESEARCH INTEREST:

Second order linear and nonlinear partial differential equations with non- negative characteristic form and applications. Harmonic Analysis in sub- riemannian settings.

—PUBLICATIONS: About 30 papers in refereed journals.

3) ANDREA BONFIGLIOLI:

—Education and Undergraduate Studies: July 1998, Universita' di Bologna (Mathematics)

—Career/Employment:

March 2002: Ph.D. in Mathematical Analysis at Dipartimento di Matematica, Universita' di Bologna (Italy).

November 2006: Assistant Professor at Dipartimento di Matematica, Universita' di Bologna.

—CURRENT RESEARCH INTEREST:

Second order linear partial differential equations with non-negative characteristic form and applications. Potential Theory in stratified Lie groups.

—PUBLICATIONS: About 20 papers in refereed journals.

Table of Contents

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.

Editorial Reviews

From the reviews:

"The book is about sub-Laplacians on stratified Lie groups. The authors present the material using an elementary approach. They achieve the level of current research starting from the basic notions of differential geometry and Lie group theory. The book is full of extensive examples which illustrate the general problems and results. Exercises are included at the end of each chapter. … The book is clearly and carefully written. It will be useful for both the graduate student and researchers in different areas." (Roman Urban, Zentralblatt MATH, Vol. 1128 (6), 2008)

"The monograph under review is a comprehensive treatment of many interesting results regarding subelliptic partial differential equations. The first aim of this book is to give a complete overview on stratified Lie groups and their Lie algebras of left-invariant vector fields. … addressed to specialists in this area." (Maria Stella Fanciullo, Mathematical Reviews, Issue 2009 m)

From the Publisher