Nested nodal loops of biharmonic functions

TL;DR

This paper constructs biharmonic polynomials in two variables with zero sets containing nested loops, revealing insights related to classical conjectures and expanding understanding of biharmonic functions.

Contribution

It provides explicit constructions of nested nodal loops in biharmonic polynomials, addressing longstanding questions in the theory of biharmonic functions.

Findings

Constructed biharmonic polynomials with arbitrary nested loops

Connected the case of two nested loops to the Boggio-Hadamard conjecture

Extended understanding of zero sets of biharmonic functions

Abstract

Given any \(n\in\mathbb{N}\), we construct a real-valued biharmonic polynomial on \(\mathbb{R}^2\) whose zero set contains a nest of \(n\) smooth, disjoint topological loops, meaning that the \(k\)-th loop lies inside the domain bounded by the \((k+1)\)-st loop for \(k=1,\ldots,n-1\). The case \(n=2\), i.e., the existence of two nested loops, is related to the failure of the Boggio-Hadamard conjecture from the early 1900s.