On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

Abstract

We construct two-term asymptotics $$\lambda ^\varepsilon _k= \varepsilon ^{m-2}(M+\varepsilon \mu _k+ O(\varepsilon ^{3/2}) )$$ of eigenvalues of a mixed boundary-value problem in $$\Omega \subset {{\mathbb {R}}}^2$$ with many heavy ( $$m>2$$ ) concentrated masses near a straight part $$\Gamma $$ of the boundary $$\partial \Omega $$ . $$\varepsilon $$ is a small positive parameter related to size and periodicity of the masses; $$k\in {\mathbb N}$$ . The main term $$M>0$$ is common for all eigenvalues but the correction terms $$\mu _k$$ , which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on $$\Gamma $$ , exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a “strongly” singular weight.