Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications

Abstract

CONTENTS Introduction Chapter I. (Subsidiary). Fundamental solutions of second-order parabolic equations § 1. Fundamental solutions § 2. The Cauchy problem Chapter II. Harnack's inequality § 1. Proof of Harnack's inequality § 2. Consequences of Harnack's inequality Chapter III. Two-sided estimates of classical fundamental solutions § 1. Preliminary estimates § 2. An upper estimate of the fundamental solution § 3. A lower estimate of the fundamental solution Appendix Chapter IV. Estimates of the derivatives of classical fundamental solutions of stationary equations § 1. Estimates of the time derivatives of fundamental solutions § 2. Integral estimates of special differential expressions of fundamental solutions Chapter V. Weak fundamental solutions of parabolic and elliptic equations with measurable coefficients § 1. Definitions. Energy inequalities § 2. Weak fundamental solutions of the Cauchy problem § 3. Weak fundamental solutions of stationary parabolic and elliptic equations Chapter VI. Generalizations. Applications § 1. Generalization to the case of equations containing lower-order derivatives § 2. The asymptotic proximity of solutions of the Cauchy problem. A stabilization theorem Comments References