## Neumann to Steklov eigenvalues: asymptotic and monotonicity results.(English)Zbl 1372.35193

Summary: We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.

### MSC:

 35P05 General topics in linear spectral theory for PDEs 35C20 Asymptotic expansions of solutions to PDEs 35P15 Estimates of eigenvalues in context of PDEs 35B25 Singular perturbations in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$
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