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      Parabolic theory as a high-dimensional limit of elliptic theory

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          Abstract

          The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows. Specifically, we give new proofs of an \(L^2\) Carleman estimate for the heat operator, and the monotonicity formulas for the frequency function associated to the heat operator, the two-phase free boundary problem, the flow of harmonic maps, and the mean curvature flow. The proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory. In particular, each parabolic theorem is proved by taking a high-dimensional limit of the related elliptic result.

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          Author and article information

          Journal
          2014-11-05
          Article
          1411.1333

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          Custom metadata
          35J15, 35K10
          math.AP

          Analysis

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