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Mathematische Annalen

, Volume 341, Issue 2, pp 429–454 | Cite as

A parabolic almost monotonicity formula

  • Anders Edquist
  • Arshak Petrosyan
Article

Abstract

We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions u ±(x, s) in the strip \(S_1 = {\mathbb{R}}^n \times (-1, 0]\) satisfying
$$u_\pm \geq 0,\quad (\Delta - \partial_s)u_\pm \geq -1, \quad u_+\cdot u_- = 0 \quad {\rm in}S_1.$$
We also establish a localized version of the formula as well as prove one of its variants. At the end of the paper we give an application to a free boundary problem related to the caloric continuation of heat potentials.

Mathematics Subject Classification (2000)

Primary 35K10 Secondary 35R35 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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