A self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula
Emanuele Salato
TL;DR
This paper offers a self-contained proof of the Alt-Caffarelli-Friedman monotonicity formula, a fundamental result in free boundary problems, utilizing convexity properties to establish key inequalities.
Contribution
It provides a new, self-contained proof of the formula, simplifying the understanding of its core components and the Friedland-Hayman inequality.
Findings
Proof of the Alt-Caffarelli-Friedman monotonicity formula
Establishment of the Friedland-Hayman inequality using convexity
Clarification of foundational aspects in free boundary theory
Abstract
The Alt-Caffarelli-Friedman monotonicity formula is a cornerstone in the theory of free boundary problems. In this note we provide a self-contained proof of this result. To prove the main stepping stone, namely the Friedland-Hayman inequality, we exploit a useful convexity property.
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Taxonomy
TopicsMathematical Inequalities and Applications · Matrix Theory and Algorithms · Functional Equations Stability Results