Singularities at the contact point of two kissing Neumann balls

We investigate eigenfunctions of the Neumann Laplacian in a bounded domain Ω⊂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 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 Rd−1∖O. We also discuss other shapes producing thinning gaps between touching cavities.