On a Weiss-type Almost Monotonicity Formula

TL;DR

This paper develops a Weiss-type almost-monotonicity formula for variable-coefficient energy functionals, enabling classification of blow-up limits and extending free-boundary regularity results under minimal regularity assumptions.

Contribution

It introduces a new almost-monotonicity formula applicable to broad classes of variable-coefficient problems with minimal regularity, advancing free-boundary analysis.

Findings

Classified blow-up limits for the Alt--Phillips problem with weaker regularity assumptions.

Extended free-boundary regularity results using a novel argument.

Discussed potential extensions including two-phase problems.

Abstract

We establish a Weiss-type almost-monotonicity formula for a broad class of variable-coefficient energy functionals, assuming only minimal regularity of the coefficients. As an application, we classify blow-up limits for the Alt--Phillips problem with variable coefficients under significantly weaker regularity hypotheses than those imposed in Ara\'ujo et al. [Calc. Var. Partial Differential Equations, 65, no.~1, Paper No.~24 (2026)]. Moreover, by means of a distinct argument, we extend the corresponding free-boundary regularity result. We conclude with a discussion of further extensions, including two-phase analogues.