A note on parabolic power concavity

Abstract

We investigate parabolic power concavity properties of the solutions of the heat equation in W Â ½0; TÞ, where W ¼ R n or W is a bounded convex domain in R n .

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 .…”

Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems

“…Reults similar to ours have been shown in [BS13] for very general operators and in [Kul17] for the fractional Laplacian ∆ 1 2 in the planar case. Power concavity for parabolic equations was discussed in [IS14] and [Zha17]. To prove the desired power concavity, we use the comparison principle [LW08b, Theorem 1.3, Theorem 2.4] and show that the convex envelope of the function −u α+1 α+2 coincides with the function itself.…”

Existence and convexity of solutions of the fractional heat equation

“…We also address the study of geometric properties of the solution of problem (1.2), focusing on convexity in the space variable x (see Section 4). In the classical case s = 1 we know that the heat flow preserves convexity: quoting [18], "it is easily seen that, when Ω = R N , every solution of [the heat equation] with moderate growth at space infinity preserves the spatial concavity of [the initial datum] at any time t > 0". We prove that a similar result holds in the fractional case s ∈ (1/2, 1), and give precise statements in Theorem 4.7 and Theorem 4.8.…”