Doubling inequality and nodal sets for solutions of bi-Laplace equations

TL;DR

This paper studies the properties of solutions to bi-Laplace equations, establishing bounds on their nodal sets and doubling inequalities, which inform about their vanishing behavior and geometric structure.

Contribution

It extends recent developments on Laplace eigenfunctions to bi-Laplace equations, providing new bounds and inequalities for their nodal sets and vanishing rates.

Findings

Polynomial upper bound for nodal sets of solutions and gradients

Implicit upper bound for nodal sets of solutions

Two types of doubling inequalities for solutions

Abstract

We investigate the doubling inequality and nodal sets for the solutions of bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper bound for the nodal sets of solutions. We show two types of doubling inequalities for the solutions of bi-Laplace equations. As a consequence, the rate of vanishing is given for the solutions.