Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Measure Upper Bounds for Nodal Sets of Eigenfunctions of the bi-Harmonic
  Operator   In this article, we consider eigenfunctions $u$ of the bi-harmonic operator,
i.e.,
  $\triangle^2u=\lambda^2u$ on $\Omega$ with some homogeneous linear boundary
conditions. We assume that $\Omega\subseteq\mathbb{R}^n$ ($n\geq2$) is a
$C^{\infty}$ bounded domain, $\partial\Omega$ is piecewise analytic and
$\partial\Omega$ is analytic except a set $\Gamma\subseteq\partial\Omega$ which
is a finite union of some compact $(n-2)$ dimensional submanifolds of
$\partial\Omega$. The main result of this paper is that the measure upper
bounds of the nodal sets of the eigenfunctions is controlled by
$\sqrt{\lambda}$. We first define a frequency function and a doubling index
related to these eigenfunctions. With the help of establishing the monotonicity
formula, doubling conditions and various a priori estimates, we obtain that the
$(n-1)$ dimensional Hausdorff measures of nodal sets of these eigenfunctions in
a ball are controlled by the frequency function and $\sqrt{\lambda}$. In order
to further control the frequency function with $\sqrt{\lambda}$, we first
establish the relationship between the frequency function and the doubling
index, and then separate the domain $\Omega$ into two parts: a domain away from
$\Gamma$ and a domain near $\Gamma$, and develop iteration arguments to deal
with the two cases respectively.

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The categories of this paper are: hep-th
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The categories of this paper are: econ
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph