Cinti, Chiara
(2008)
*Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators.*
[Preprint]

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## Abstract

We consider a class of hypoelliptic ultraparabolic operators in the form L = (X_1)^2 + ... + (X_m)^2 + X_0 - \partial_t, under the assumption that the vector fields X_1, ..., X_m and X_0-\partial_t are invariant with respect to a suitable homogeneous Lie group G. We show that if u,v are two solutions of Lu = 0 on R^Nx]0,T[ and u(x,0)=\phi, then each of the following conditions: |u(x,t)-v(x,t)| can be bounded by M exp(c(|x|_G)^2), or both u and v are non negative, implies u=v. We use a technique which relies on a pointwise estimate of the fundamental solution of L.

Abstract

We consider a class of hypoelliptic ultraparabolic operators in the form L = (X_1)^2 + ... + (X_m)^2 + X_0 - \partial_t, under the assumption that the vector fields X_1, ..., X_m and X_0-\partial_t are invariant with respect to a suitable homogeneous Lie group G. We show that if u,v are two solutions of Lu = 0 on R^Nx]0,T[ and u(x,0)=\phi, then each of the following conditions: |u(x,t)-v(x,t)| can be bounded by M exp(c(|x|_G)^2), or both u and v are non negative, implies u=v. We use a technique which relies on a pointwise estimate of the fundamental solution of L.

Document type

Preprint

Creators

Keywords

H\"{o}rmander operators, ultraparabolic operators, Cauchy problem, uniqueness theorems, homogeneous Lie groups

Subjects

DOI

Deposit date

04 Nov 2008

Last modified

16 May 2011 12:09

URI

## Other metadata

Document type

Preprint

Creators

Keywords

H\"{o}rmander operators, ultraparabolic operators, Cauchy problem, uniqueness theorems, homogeneous Lie groups

Subjects

DOI

Deposit date

04 Nov 2008

Last modified

16 May 2011 12:09

URI

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