JournalsrlmVol. 20, No. 2pp. 145–158

Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators

  • Chiara Cinti

    Università di Bologna, Italy
Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators cover
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Abstract

We consider a class of hypoelliptic ultraparabolic operators in the form

\L=j=1mXj2+X0t,\L=\sum_{j=1}^m X_j^2+X_0-\partial_t,

under the assumption that the vector fields X1,,XmX_1,\ldots,X_m and X0tX_0-\partial_t are invariant with respect to a suitable homogeneous Lie group G\mathbb{G}. We show that if u,vu,v are two solutions of \Lu=0\L u = 0 on \RN×]0,T[\RN \times ]0,T[ and u(,0)=φu(\cdot,0)=\varphi, then each of the following conditions: u(x,t)v(x,t)|u(x,t)-v(x,t)| can be bounded by Mexp(cxG2)M \exp (c|x|_{\mathbb{G}}^2), or both uu and vv are non negative, implies uvu\equiv v. We use a technique which relies on a pointwise estimate of the fundamental solution of \L\L.

Cite this article

Chiara Cinti, Uniqueness in the Cauchy problem for a class of hypoelliptic ultraparabolic operators. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 20 (2009), no. 2, pp. 145–158

DOI 10.4171/RLM/538