Regular solutions of transmission and interaction problems for wave equations

Consider n bounded domains Ω ⊆ ℝ   k   iand elliptic formally symmetric differential operators A 1 of second order on Ω i Choose any closed subspace V in \documentclass{article}\pagestyle{empty}\begin{document}$ \prod\limits_{i = 1}^n {L^2 \left({\Omega _i } \right)} $\end{document} , and extend ( A i ) i =1,…, n by Friedrich's theorem to a self‐adjoint operator A with D ( A 1/2 ) = V (interaction operator). We give asymptotic estimates for the eigenvalues of A and consider wave equations with interaction. With this concept, we solve a large class of problems including interface problems and transmission problems on ramified spaces. 25,32 We also treat non‐linear interaction, using a theorem of Minty 29 .