[Paper Review] Counterexamples to quasiconcavity for the heat equation

Albert Chau, Ben Weinkove|arXiv (Cornell University)|Feb 13, 2018

Geometric Analysis and Curvature Flows24 references3 citations

TL;DR

This paper constructs explicit counterexamples to quasiconcavity preservation under the heat equation on convex annular domains, demonstrating that even smooth, subharmonic, and quasiconcave initial data can evolve into solutions whose spatial superlevel sets lose convexity over time. The authors use perturbations of radial heat flows with non-spherical level sets to break quasiconcavity, and further show via a two-point function argument that space-time quasiconcavity fails even under radial symmetry.

ABSTRACT

We construct solutions to the heat equation on convex rings showing that quasiconcavity may not be preserved along the flow, even for smooth and subharmonic initial data.

Motivation & Objective

• To investigate whether quasiconcavity of initial data is sufficient to preserve spatial or space-time quasiconcavity under the heat flow.
• To challenge the assumption that subharmonicity of initial data ensures quasiconcavity preservation in parabolic flows.
• To construct explicit counterexamples in dimensions $ n \geq 2 $ where superlevel sets of the solution become non-convex after positive time.
• To demonstrate that radial symmetry of initial data does not guarantee space-time quasiconcavity of the solution.
• To analyze the failure of the two-point function $ \mathcal{H} $ to remain non-positive, indicating breakdown of space-time quasiconcavity.

Proposed method

• Construct a radial initial function $ V $ with rapid decay from 1 to 0, ensuring level sets are spheres under heat flow.
• Design a non-radial initial function $ W $ whose level sets are ellipsoids near $ \Omega_1 $, transitioning to spherical shapes outward.
• Form a convex combination $ u_0 = (1-\varepsilon)V + \varepsilon W $ for small $ \varepsilon > 0 $, preserving admissibility and quasiconcavity of initial data.
• Use the heat flow to evolve $ u_0 $, showing that interaction between spherical and ellipsoidal level sets causes non-convex superlevel sets at positive time.
• Apply a two-point function $ \mathcal{H}((x,s),(y,t)) = (Du(y,t)-Du(x,s))\cdot(y-x) + (u_t(y,t)-u_t(x,s))(t-s) $ to detect failure of space-time quasiconcavity.
• Analyze $ \mathcal{H} $ at $ s=0 $, showing it becomes positive for large $ t $, contradicting the necessary condition for space-time quasiconcavity.

Experimental results

Research questions

• RQ1 Does quasiconcavity of initial data imply that $ x \mapsto u(x,t) $ remains quasiconcave for all $ t > 0 $?
• RQ2 Can subharmonic initial data with compact support preserve quasiconcavity under the heat equation?
• RQ3 Is space-time quasiconcavity preserved when the initial data is radial and quasiconcave?
• RQ4 Does the two-point function $ \mathcal{H} $ remain non-positive for solutions of the heat equation on convex rings?
• RQ5 What is the minimal regularity and geometric condition on initial data that ensures quasiconcavity preservation under the heat flow?

Key findings

• For $ n \geq 2 $, there exists a smooth, quasiconcave, subharmonic initial function $ u_0 $ on the annulus $ 1 < r < 2 $ such that $ u(x,t_0) $ fails to be quasiconcave for some $ t_0 > 0 $.
• The solution $ u(x,t) $ to the heat equation starting from such $ u_0 $ develops non-convex superlevel sets in space, violating spatial quasiconcavity after positive time.
• Even when $ u_0 $ is radially symmetric and quasiconcave, the solution $ u(x,t) $ fails to be space-time quasiconcave, as shown by a positive two-point function $ \mathcal{H} $ at large $ t $.
• The counterexample relies on the non-convexity of the union of a sphere and a non-spherical ellipsoid, exploited via perturbation of radial heat flow.
• The two-point function $ \mathcal{H} $ is shown to be strictly positive at certain points $ ((x,0),(y,t)) $, contradicting the necessary condition for space-time quasiconcavity.
• The construction invalidates a claim in [15, Theorem 3] and highlights the need for stronger hypotheses in parabolic quasiconcavity results.