A Quasiconcavity Property for the Heat Equation in a Convex Ring

Abstract

We give an exposition of a result of Borell (Commun Math Phys 86:143–147, 1982) that the probability function that Brownian motion hits the inner boundary before time t and before hitting the outer boundary is a space-time quasiconcave function.

Keywords

Heat equation / Quasiconcavity / Maximum principle / Comparison method

References

[1] Borell C. Brownian motion in a convex ring and quasi-concavity. Commun. Math. Phys.. 1982, 86 1 143-147

[2] Chau, A., Weinkove, B.: Counterexamples to the quasiconcavity for heat equation, preprint arXiv:1802.04770

[3] Diaz JI, Kawohl B. On convexity and starshapedness of level sets for some non-linear elliptic and parabolic problems on convex rings. J. Math. Anal. Appl.. 1993, 177 263-286

[4] Gabriel R. A result concerning convex level surface of 3-dimensional harmonic functions. J. Lond. Math. Soc.. 1957, 32 286-294

[5] Lewis JL. Capacitary function in convex rings. Arch. Ration. Mech. Anal.. 1977, 66 201-224

Funding

Chinese Government Scholarship(201606340046)