2024/07/09 by Josef Greilhuber, Greilhuber, Josef Eberhard · 1 citation
Mathematics · #33C50 #33C55 #35J15 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical functions and polynomials #Primary: 31B05 #Secondary: 35J05
paper · pdf · doi:10.48550/arxiv.2407.07039
openalex publication_date 2024/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that, in dimension three and higher, the space of harmonic functions vanishing on the cone defined by a generically chosen harmonic quadratic polynomial is two-dimensional. This phenomenon is surprisingly robust, generalizing to arbitrary elliptic differential operators of second order, with the cone replaced by the level set of a solution at a nondegenerate critical value. As long as the tangent cone to the level set at the critical point satisfies a certain genericity condition, the space of solutions vanishing on the level set is at most two-dimensional.