On Space-Time Quasiconcave Solutions of the Heat Equation.

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.

In this paper the authors 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, they 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 their ideas and for completeness, the authors also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.