On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions
Let \(\Omega\subset \mathbb{R}^N\) be a bounded connected open set, \(A\) a linear elliptic differential operator of order \(\mu\), \(\ell\) a fixed natural number between 1 and \(\mu/2\), \(B_j u\), \(j= 1,\dots, \ell\) and \(C_k u\), \(k= 1,\dots, \mu/2\) linear boundary operators. The authors study eigenvalue problems of the form: \[ Au= \lambda u\text{ in } \Omega,\;B_j u= \lambda C_j u,\;j= 1,\dots,\ell,\;C_k u= 0,\;k= \ell+ 1,\dots, \mu/2,\text{ on } \partial \Omega.\tag{1} \] \(\lambda_0\) is called an eigenvalue of problem (1), if (1) with \(\lambda_0\) has at least one non-trivial solution \(u_0\in H^\mu(\Omega)\). Let \(\lambda_0\) and \(u_0\) be an eigenvalue resp. eigenfunction. Consider the problems: \[ (A- \lambda_0 I) u_p= u_{p- 1}\text{ in } \Omega,\;(B_j- \lambda_0 C_j) u_p= C_j u_{p- 1},\;j= 1,\dots, \ell,\tag{2} \] \[ C_k u_p= 0,\;k= \ell+ 1,\dots, \mu/2\text{ on } \partial \Omega,\;p= 1,2,\dots\;. \] A solution of problem (2) is called an associated function of the \(p\)th rank to the eigenfunction \(u_0\). Eigenfunctions and associated functions of problem (1) are combined under the general name root functions. The authors establish completeness and summability properties of the root functions.
scientific article; zbMATH DE number 811613