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Critical semilinear equations on the Heisenberg group

Critical semilinear equations on the Heisenberg group Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Suppl. Vol. XLV, 187-195 (1999) E. LANCONELLI (*) To the memory of Lamberto Cattabriga 1. - Introduction. In this note we report on a series of results by G. Citti, F. Uguzzoni and the present author concerning the Dirichlet boundary value problem for semilin- ear equation with critical behavior on the Heisenberg group. We begin by recalling some definitions and some basic known results. The Heisenberg group H ~, whose points will be denoted by ~ = (z, t) = (x, y, t), is the Lie group (R 2~ +1, o) with composition law defined by ~..,~' = (z +z', t +t' +2((x', y)- (x, y'))) where (,} denotes the inner product in R" . As is well known, the linear second order partial differential operator j=l where (1) Xj=~xj+2yj~t, Yj=3yj-2xj~t, je{1, ..., n}, is called the Kohn Laplacian on H'. The operator A H" has nonnegative defined characteristic form with mini- mum eigenvalue identically zero. Then A Hn is degenerate elliptic and not ellip- tic at any point. However, due to a well known theorem of HSrmander, the Kohn Laplacian is a sub-elliptic hypoelliptic operator since the Lie algebra http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL UNIVERSITA DI FERRARA Springer Journals

Critical semilinear equations on the Heisenberg group

Lanconelli, E.
ANNALI DELL UNIVERSITA DI FERRARA , Volume 45 (1) – Jan 1, 1999
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References (13)

Publisher
Springer Journals
Copyright
Copyright © Università degli Studi di Ferrara 1999
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/bf02826094
Publisher site
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Abstract

Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Suppl. Vol. XLV, 187-195 (1999) E. LANCONELLI (*) To the memory of Lamberto Cattabriga 1. - Introduction. In this note we report on a series of results by G. Citti, F. Uguzzoni and the present author concerning the Dirichlet boundary value problem for semilin- ear equation with critical behavior on the Heisenberg group. We begin by recalling some definitions and some basic known results. The Heisenberg group H ~, whose points will be denoted by ~ = (z, t) = (x, y, t), is the Lie group (R 2~ +1, o) with composition law defined by ~..,~' = (z +z', t +t' +2((x', y)- (x, y'))) where (,} denotes the inner product in R" . As is well known, the linear second order partial differential operator j=l where (1) Xj=~xj+2yj~t, Yj=3yj-2xj~t, je{1, ..., n}, is called the Kohn Laplacian on H'. The operator A H" has nonnegative defined characteristic form with mini- mum eigenvalue identically zero. Then A Hn is degenerate elliptic and not ellip- tic at any point. However, due to a well known theorem of HSrmander, the Kohn Laplacian is a sub-elliptic hypoelliptic operator since the Lie algebra

Journal

ANNALI DELL UNIVERSITA DI FERRARA – Springer Journals

Published: Jan 1, 1999

Keywords: Critical Exponent; Heisenberg Group; Nonnegative Solution; Nonlinear Elliptic Equation; Yamabe Problem

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