A Quasiconcavity Property for the Heat Equation in a Convex Ring

Jingjing Suo

Communications in Mathematics and Statistics ›› 2020, Vol. 8 ›› Issue (4) : 453 -462.

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Communications in Mathematics and Statistics ›› 2020, Vol. 8 ›› Issue (4) : 453 -462. DOI: 10.1007/s40304-020-00207-6
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A Quasiconcavity Property for the Heat Equation in a Convex Ring

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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

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Jingjing Suo. A Quasiconcavity Property for the Heat Equation in a Convex Ring. Communications in Mathematics and Statistics, 2020, 8(4): 453-462 DOI:10.1007/s40304-020-00207-6

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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)

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