Upper bounds of nodal sets for solutions of bi-Laplace equations: II

TL;DR

This paper establishes polynomial upper bounds for the size of nodal sets of solutions to bi-Laplace equations using Carleman estimates, avoiding the traditional frequency function approach.

Contribution

It introduces a new method based on Carleman estimates to bound nodal sets, bypassing the need for frequency functions in bi-Laplace problems.

Findings

Polynomial upper bounds for nodal sets achieved

Monotonicity and propagation of smallness results derived

Method avoids reliance on frequency functions

Abstract

We investigate the upper bounds of nodal sets for solutions of bi-Laplace equations without using frequency functions which play an essential role in the study of nodal sets in the celebrated work by Logunov \cite{Lo18}. We obtain some delicate monotonicity and propagation of smallness results by Carleman estimates. A polynomial upper bound for the nodal sets of solutions is obtained.