On Space-Time Quasiconcave Solutions of the Heat Equation

Abstract

In this paper we first obtain a constant rank theorem for the second fundamental form of the spacetime 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 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.

The microscopic convexity of level sets of solutions for elliptic and parabolic equations

“…抛物方程时空凸解的常秩定理首先由文献 [28,29] 得到, 它们在时空凸水平集的第二基本形式的 常秩定理研究中起到重要作用. Chen 等 [25] 建立解的时空水平集的常秩定理如下: 定理 3.2 [25] 假设…”

Convexity for free boundaries with singular term (nonlinear elliptic case)

“…Convexity configurations in parabolic partial differential equations have been extensively explored in the literature; refer to Definition 1 for various convexity concepts. Noteworthy studies include parabolic quasiconcavity in [17,18], space-time quasiconcavity in [6,9], and a more general notion of parabolic power concavity in [19,20]. The concept of a "spatially quasiconvex envelope" in the parabolic setting is discussed in [22].…”

“…The concept of a "spatially quasiconvex envelope" in the parabolic setting is discussed in [22]. Additional references encompass [7,9,12,17,18,19,20,22,23,24].…”