The uniqueness and existence of solutions for the scattering problem for a fully coated penetrable cavity are studied by using variational method. First

the differential equations are transformed into an integral equation by using the Greens first identity.  Then from the Rellichs lemma and a unique continuation principle

the uniqueness of solutions for the scattering problem for a fully coated penetrable cavity is proven. Next

from the Dirichlet-to-Neumann operator

the trace theorem

the continuous embedding theorem and the Lax-Milgrams lemma

the existence of solutions for the scattering problem for a fully coated penetrable cavity is also proven.