Abstract

This talk explores the asymptotic properties of harmonic functions near conical boundary points, addressing three primary questions: the conditions under which harmonic functions are asymptotically homogeneous, the uniqueness of Almgren blow-up limits, and the existence of asymptotic expansions. Unlike previous studies focused on domains with at least $C^{1,Dini}$ smoothness, this work considers domains meeting specific one-point conditions where traditional Almgren-type monotonicity formulas do not apply. To address these challenges, we introduce a compactness method as an alternative tool. This is joint work with Dennis Kriventsov (Rutgers).