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Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities
About this Title
Marco Bramanti, Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy, Luca Brandolini, Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università di Bergamo, Viale Marconi 5, 24044 Dalmine, Italy, Ermanno Lanconelli, Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy and Francesco Uguzzoni, Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 204, Number 961
ISBNs: 978-0-8218-4903-3 (print); 978-1-4704-0575-5 (online)
DOI: https://doi.org/10.1090/S0065-9266-09-00605-X
Published electronically: November 9, 2009
Keywords: Hörmander’s vector fields,
heat kernels,
Gaussian bounds,
Harnack inequalities
MSC: Primary 35H20, 35A08, 35K65; Secondary 35H10, 35A17
Table of Contents
Chapters
- Introduction
- Part I: Operators with constant coefficients
- Part II: Fundamental solution for operators with Hölder continuous coefficients
- Part III: Harnack inequality for operators with Hölder continuous coefficients
- Epilogue
Abstract
In this work we deal with linear second order partial differential operators of the following type: \[ H=\partial _{t}-L=\partial _{t}-\sum _{i,j=1}^{q}a_{ij}\left ( t,x\right ) X_{i}X_{j}-\sum _{k=1}^{q}a_{k}\left ( t,x\right ) X_{k}-a_{0}\left ( t,x\right ) \] where $X_{1},X_{2},\ldots ,X_{q}$ is a system of real Hörmander’s vector fields in some bounded domain $\Omega \subseteq \mathbb {R}^{n}$, $A=\left \{ a_{ij}\left ( t,x\right ) \right \} _{i,j=1}^{q}$ is a real symmetric uniformly positive definite matrix such that: \[ \lambda ^{-1}\left \vert \xi \right \vert ^{2}\leq \sum _{i,j=1}^{q}a_{ij}\left ( t,x\right ) \xi _{i}\xi _{j}\leq \lambda \left \vert \xi \right \vert ^{2}\text { }\forall \xi \in \mathbb {R}^{q},x\in \Omega ,t\in \left ( T_{1},T_{2}\right ) \] for a suitable constant $\lambda >0$ a for some real numbers $T_{1}<T_{2}.$ The coefficients $a_{ij},a_{k},a_{0}$ are Hölder continuous on $\left ( T_{1},T_{2}\right ) \times \Omega$ with respect to the parabolic CC-metric \[ d_{P}\left ( \left ( t,x\right ) ,\left ( s,y\right ) \right ) =\sqrt {d\left ( x,y\right ) ^{2}+\left \vert t-s\right \vert }\] (where $d$ is the Carnot-Carathéodory distance induced by the vector fields $X_{i}$’s). We prove the existence of a fundamental solution $h\left ( t,x;s,y\right )$ for $H$, satisfying natural properties and sharp Gaussian bounds of the kind:\begin{align*} \frac {e^{-cd\left ( x,y\right ) ^{2}/\left ( t-s\right ) }}{c\left \vert B\left ( x,\sqrt {t-s}\right ) \right \vert }&\leq h\left ( t,x;s,y\right ) \leq c\frac {e^{-d\left ( x,y\right ) ^{2}/c\left ( t-s\right ) }}{\left \vert B\left ( x,\sqrt {t-s}\right ) \right \vert }\\ \left \vert X_{i}h\left ( t,x;s,y\right ) \right \vert &\leq \frac {c}{\sqrt {t-s}}\frac {e^{-d\left ( x,y\right ) ^{2}/c\left ( t-s\right ) }}{\left \vert B\left ( x,\sqrt {t-s}\right ) \right \vert }\\ \left \vert X_{i}X_{j}h\left ( t,x;s,y\right ) \right \vert +\left \vert \partial _{t}h\left ( t,x;s,y\right ) \right \vert &\leq \frac {c}{t-s}\frac {e^{-d\left ( x,y\right ) ^{2}/c\left ( t-s\right ) }}{\left \vert B\left ( x,\sqrt {t-s}\right ) \right \vert }\end{align*} where $\left \vert B\left ( x,r\right ) \right \vert$ denotes the Lebesgue measure of the $d$-ball $B\left ( x,r\right )$. We then use these properties of $h$ as a starting point to prove a scaling invariant Harnack inequality for positive solutions to $Hu=0$, when $a_{0}\equiv 0$. All the constants in our estimates and inequalities will depend on the coefficients $a_{ij},a_{k},a_{0}$ only through their Hölder norms and the number $\lambda$.- D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890–896. MR 217444, DOI 10.1090/S0002-9904-1967-11830-5
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