TY - JOUR

T1 - Singularities at the contact point of two kissing Neumann balls

AU - Nazarov, S.A.

AU - Taskinen, J.

PY - 2018/2/5

Y1 - 2018/2/5

N2 - We investigate eigenfunctions of the Neumann Laplacian in a bounded domain Ω⊂R d, 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 Ω‾, if the dimension d equals 2, but in dimension d≥3 their gradients have a strong singularity O(|x−O| −α), α∈(0,2−2] 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 R d−1∖O. We also discuss other shapes producing thinning gaps between touching cavities.

AB - We investigate eigenfunctions of the Neumann Laplacian in a bounded domain Ω⊂R d, 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 Ω‾, if the dimension d equals 2, but in dimension d≥3 their gradients have a strong singularity O(|x−O| −α), α∈(0,2−2] 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 R d−1∖O. We also discuss other shapes producing thinning gaps between touching cavities.

KW - Asymptotic analysis

KW - Boundary singularity

KW - Eigenfunction

KW - Kondratiev theory

KW - Laplace–Neumann problem

KW - Tangential balls

UR - http://www.scopus.com/inward/record.url?scp=85031093617&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2017.09.044

DO - 10.1016/j.jde.2017.09.044

M3 - Article

VL - 264

SP - 1521

EP - 1549

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 3

ER -