Regular solutions of transmission and interaction problems for wave equations

Consider n bounded domains Ω ? ? and elliptic formally symmetric differential operators A₁ of second order on Ω_i Choose any closed subspace V in $ \prod\limits_{i = 1}^n {L^2 \left({\Omega _i } \right)} $, 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 Minty²⁹.