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

Abstract

This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results are applicable to the Pucci operator, the normalized q-Laplacians with 1 < q ≤ ∞, the Finsler Laplacian, and more general quasilinear operators.

Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations

“…We show a key ingredient to prove (4.1), which stems from the idea in [2] to prove convexity of solutions to fully nonlinear equations by using its convex envelope. Such an idea is later developed in [20] to show a power-type convexity or concavity with a finite exponent. We here makes a further step, studying the limit case as the exponent tends to ∞.…”

“…A non-exhaustive list of references includes [32,29,30,7,6] for classical solutions and [17,2,27,33] for viscosity solutions. A generalized type, called power convexity/concavity, is investigated in [23,24,20,25] for various equations. One major idea applied in [29,2,20] is to show the corresponding convex envelope of a solution is a supersolution of the equation and then use the comparison principle to conclude the proof.…”

“…A generalized type, called power convexity/concavity, is investigated in [23,24,20,25] for various equations. One major idea applied in [29,2,20] is to show the corresponding convex envelope of a solution is a supersolution of the equation and then use the comparison principle to conclude the proof. This machinery can be implemented also for quasiconvexity or quesiconcavity of classical solutions to the Dirichlet boundary problems for semilinear elliptic or parabolic equations [15,13,22].…”

Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

“…Concavity of solutions to elliptic and parabolic boundary value problems on convex domains in Euclidean space is a classical subject and has fascinated many mathematicians. The literature is large and we just refer to the classical monograph by Kawohl [19] and the papers [3], [6], [10], [12]- [18], [20], [21], [23] and some of which are closely related to this paper and the others include recent developments in this subject. In this regard, let us recall some results on power concavity properties of solutions to elliptic and parabolic boundary value problems.…”

Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

“…Such a supersolution preserving property enables us to obtain the convexity of the solution immediately if the comparison principle for the equation is known to hold. We refer the reader also to [22] and recent work [20,10] for more applications of this method in the Euclidean space.…”