Electronic ISBN:  9781470405755 
Product Code:  MEMO/204/961.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 204; 2009; 123 ppMSC: Primary 35;
In this work the authors deal with linear second order partial differential operators of the following type \[ H=\partial_{t}L=\partial_{t}\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}\sum_{k=1}^{q}a_{k}(t,x)X_{k}a_{0}(t,x)\] 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 \[\lambda^{1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})\] for a suitable constant \(\lambda>0\) a for some real numbers \(T_{1} < T_{2}\).

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


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In this work the authors deal with linear second order partial differential operators of the following type \[ H=\partial_{t}L=\partial_{t}\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}\sum_{k=1}^{q}a_{k}(t,x)X_{k}a_{0}(t,x)\] 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 \[\lambda^{1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})\] for a suitable constant \(\lambda>0\) a for some real numbers \(T_{1} < T_{2}\).

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