Parabolic power concavity and parabolic boundary value problems

Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations

“…This type of concavity results was established in [29] and [30] (see also [27,28]). Note that (1.6) then turns into a concavity assumption for g λ .…”

“…Secondly, in accordance to our generalization of the equations, another significant contribution of this paper is that we use the weaker notion of viscosity solutions rather than the classical solutions. We thus manage to reduce the C 2 regularity of the solutions in the main theorems of [29,30]. Let us emphasize that it is indeed possible to investigate spatial convexity of solutions in the framework of viscosity theory; we refer to [17,19,1,33,40] for viscosity techniques in different contexts and to [37,38,35,6,25,26,23,24] etc for related results for classical solutions.…”

A note on parabolic power concavity

“…Inspired by [4], the authors of this paper introduced in [16] and [17] the notions of parabolic and a-parabolic quasi-concavity, and studied quasi-concavity properties involving the space and the time variables jointly for particular parabolic boundary value problems in a convex ring. Furthermore, they developed in [18] the notion of a-parabolic q-concavity for nonnegative functions and study parabolic power concavity properties of solutions to parabolic boundary value problems in a convex cylinder with vanishing initial datum and a suitable source term. See also [6,7,8,13,31] for further results related to space-time concavity of solutions of parabolic problems.…”

On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

“…In [5,6,7], Borell used the Brownian motion to study certain spacetime convexities of the solutions of diffusion equations and the level sets of the solution to a heat equations with Schrödinger potential. Ishige-Salani introduced some notions of parabolic quasiconcavity in [18,19] and parabolic power concavity in [20], which are some kinds of spacetime convexity. In [18,19,20], they studied the corresponding parabolic boundary value problems using the convex envelope method, which is a macroscopic method.…”

“…Ishige-Salani introduced some notions of parabolic quasiconcavity in [18,19] and parabolic power concavity in [20], which are some kinds of spacetime convexity. In [18,19,20], they studied the corresponding parabolic boundary value problems using the convex envelope method, which is a macroscopic method. At the same time, Hu-Ma [17] established a constant rank theorem for the space-time Hessian of space-time convex solutions to the heat equation, which is the microscopic method.…”