On a Steklov-Robin eigenvalue problem

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, consider Ω=Ω0∖B‾r, where Br is ball centered at origin with radius r>0 and Ω0⊂Rn, n⩾2, an open, bounded set Lipschitz boundary, such that B‾r⊂Ω0. We impose Steklov condition on outer boundary Robin involving positive L∞ function β(x) inner boundary. Then, first σβ(Ω) its main properties. particular, investigate behavior of when let vary L1-norm β ball. Furthermore, asymptotic corresponding eigenfunctions parameter goes to infinity.