Singularities at the contact point of two kissing Neumann balls

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http://hdl.handle.net/10138/308374

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Nazarov , S A & Taskinen , J 2018 , ' Singularities at the contact point of two kissing Neumann balls ' , Journal of Differential Equations , vol. 264 , no. 3 , pp. 1521-1549 . https://doi.org/10.1016/j.jde.2017.09.044

Title: Singularities at the contact point of two kissing Neumann balls
Author: Nazarov, Sergey A.; Taskinen, Jari
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2018-02-05
Language: eng
Number of pages: 29
Belongs to series: Journal of Differential Equations
ISSN: 0022-0396
URI: http://hdl.handle.net/10138/308374
Abstract: We investigate eigenfunctions of the Neumann Laplacian in a bounded domain Omega subset of Rd, where a cuspidal singularity is caused by a cavity consisting of two touching balls, or discs in the planar case. We prove that the eigenfunctions with all of their derivatives are bounded in (Omega)over bar, if the dimension d equals 2, but in dimension d >= 3 their gradients have a strong singularity O(vertical bar x ? O vertical bar(-alpha)), alpha is an element of (0,2 - root 2] at the point of tangency O. Our study is based on dimension reduction and other asymptotic procedures, as well as the Kondratiev theory applied to the limit differential equation in the punctured hyperplane Rd(-1) backslash O . We also discuss other shapes producing thinning gaps between touching cavities. (c) 2017 Elsevier Inc. All rights reserved.
Subject: Laplace-Neumann problem
Eigenfunction
Tangential balls
Boundary singularity
Kondratiev theory
Asymptotic analysis
SPECTRUM
BODY
WAVE
111 Mathematics
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