5
views
0
recommends
+1 Recommend
0 collections
0
shares
• Record: found
• Abstract: found
• Article: found
Is Open Access

# Parabolic theory as a high-dimensional limit of elliptic theory

Preprint

Bookmark
There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

### 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.

### Author and article information

###### Journal
2014-11-05
###### Article
1411.1333