Upper bounds of nodal sets for eigenfunctions of eigenvalue problems
- PDF / 458,554 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 96 Downloads / 249 Views
Mathematische Annalen
Upper bounds of nodal sets for eigenfunctions of eigenvalue problems Fanghua Lin1 · Jiuyi Zhu2 Received: 19 May 2020 / Revised: 29 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey–Nirenberg and Carleman estimates. Mathematics Subject Classification 35J05 · 58J50 · 35P15 · 35P20
1 Introduction The eigenvalue and eigenfunction problems are archetypical in the theory of partial differential equations. Different type of second order or higher order eigenvalue problems arise from physical phenomena in the literature. For instance, the famous Chaldni pattern is the nodal pattern modeled by the eigenfunctions of bi-Laplace eigenvalue problems. The Chladni pattern is the scientific, artistic, and even the sociological birthplace of the modern field of wave physics and quantum chaos. The goal of the paper
Communicated by Y. Giga. Lin is supported in part by NSF Grant DMS-1955249, Zhu is supported in part by NSF Grant OIA-1832961.
B
Jiuyi Zhu [email protected] Fanghua Lin [email protected]
1
Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
2
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
123
F. Lin, J. Zhu
is to provide a uniform way to obtain the upper bounds of nodal sets of eigenfunctions for various eigenvalue problems in real analytic domains. Since the nodal sets of eigenfunctions of Laplacian are well studied, we will focus on eigenfunctions of some higher order elliptic equations. The approach introduced in the paper also applies to the upper bounds of eigenfunctions of Laplacian with different boundary conditions in real analytic domains. Specifically, we consider three types of biharmonic Steklov eigenvalue problems
2 eλ = 0 λ eλ = eλ − λ ∂e ∂ν = 0
in , on ∂,
(1.1)
2 eλ = 0 2 λ eλ = ∂∂νe2λ − λ ∂e ∂ν = 0
in , on ∂
(1.2)
2 eλ = 0 ∂eλ ∂eλ 3 ∂ν = ∂ν + λ eλ = 0
in , on ∂,
(1.3)
and
where ∈ Rn with n ≥ 2 is a bounded real analytic domain, ν is a unit outer normal, and n is the dimension of the space in the paper. Those eigenvalue problems are important in biharmonic analysis, inverse problem and the theory of elasticity, see e.g. [13,22,31]. If we consider the eigenfunctions in (1.1)–(1.3) on the boundary, they become the eigenfunctions of Neumann-to-Laplacian operator, Neumann-toNeumann operator and Dirichlet to Neumann operator, respectively, see [7]. The bi-Laplace equation arises in numerous problems of structural engineering. It model
Data Loading...
