Some Quantitative Properties of Solutions to the Buckling Type Equation

TL;DR

This paper studies solutions to a Buckling type equation, establishing bounds on their nodal sets and vanishing order, and providing quantitative insights into their propagation of smallness within bounded analytic domains.

Contribution

It provides new bounds on the vanishing order and nodal set measure for solutions to the Buckling equation, along with quantitative propagation of smallness results.

Findings

Upper bounds for vanishing order depend on parameters mbda and k

Hausdorff measure of nodal sets is bounded by a constant times (mbda^{1/2} + k^{1/2} + 1)

Quantitative propagation of smallness is established

Abstract

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling type equation $\triangle^2u+\lambda\triangle u-k^2u=0$ in a bounded analytic domain $\Omega\subseteq\mathbb{R}^n$ with the homogeneous boundary conditions $u=0$ and $\frac{\partial u}{\partial\nu}=0$ on $\partial\Omega$, where $\lambda,\ k$ are nonnegative real constants, and $\nu$ is the outer unit normal vector on $\partial\Omega$. We obtain that, the upper bounds for the maximal vanishing order of $u$ and the $n-1$ dimensional Hausdorff measure of the nodal set of $u$ are both $C(\sqrt{\lambda}+\sqrt{k}+1)$, where $C$ is a positive constant only depending on $n$ and $\Omega$. Moreover, we also give a quantitative result of the propagation of smallness of $u$.