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

Lamberti, Pier Domenico; Provenzano, Luigi

Proc. R. Soc. Edinb., Sect. A, Math. 147, No. 2, 429-447 (2017).

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, \({}_0F_1\)

Keywords:

Steklov boundary conditions; eigenvalues; density perturbation; monotonicity; Bessel functions