On space-time quasiconcave solutions of the heat equation / Chuanqiang Chen, Xinan Ma, Paolo Salani.

Format:

Book

Author/Creator:

Chen, Chuanqiang (Mathematician), author.

Ma, Xinan, author.

Salani, Paolo, 1968- author.

Series:

Memoirs of the American Mathematical Society ; Volume 259, Number 1244.

Memoirs of the American Mathematical Society ; Volume 259, Number 1244

Language:

English

Subjects (All):

Heat equation--Numerical solutions.

Heat equation.

Physical Description:

1 online resource (v, 81 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2019]

Summary:

"In this Memoir we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the spacetime Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. \040Introduction

Chapter 2. Basic definitions and the Constant Rank Theorem technique

2.1. Preliminaries

2.2. A constant rank theorem for the space-time convex solution of the heat equation

2.3. The strict convexity of the level sets of harmonic functions in convex rings

Chapter 3. A microscopic space-time Convexity Principle for space-time level sets

3.1. A constant rank theorem for the spatial second fundamental form

3.2. A constant rank theorem for the space-time second fundamental form: CASE 1

3.3. A constant rank theorem for the space-time second fundamental form: CASE 2

Chapter 4. The Strict Convexity of Space-time Level Sets

4.1. The strict convexity of space-time level sets of Borell's solution

4.2. Proof of Theorem 1.0.3

Chapter 5. Appendix: the proof in dimension =2

5.1. minimal rank =0

5.2. minimal rank =1

Bibliography

Back Cover.

Notes:

Description based on print version record.

Includes bibliographical references.

ISBN:

1-4704-5243-X