The Electromagnetic Scattering Problem Of A Cavity For Maxwell's Equations

In this paper,we devote to study the electromagnetic waves scattering problem in a bounded single cavity in Euclidean space R3,the propagation of electromagnetic waves is controlled by the time-harmonic Maxwell's equations as follows:?× E-i??H=0,?× H+i??eE=0 in??2 ?R3+.Here E and H denote electric and magnetic fields.? is the angular frequency.? and ?represents the dielectric constant and permeability of the medium,and ? satisfies the condition Im ??0.The bounded open cavity ? was placed in the infinite ground,with its opening flush with the ground.Consider a time-harmonic electromagnetic plane waves from the upper part of the cavity,we can get the existence and uniqueness of the weak solution of the scattering field of the electromagnetic scattering problem in the bounded single cavity.Specifically,because we assume that above the cavity the R3+space is full of uniform media,the medium in the cavity ?is not uniform.So we can define the boundary operator T,at the cavity opening ?.The boundary operator T,maps the value of the tangential component of electric field to the tangential trace of magnetic field,thus obtaining the transparent boundary condition of the electric field E at the cavity opening,and converting the boundary value problem into the equivalent variation problem.In order to get the appropriate solution space,we need to exclude the odd solution when Im ?=0,so we define the weighted Schwartz spin space H₁/21-/2?curl;??.In order to get the existence of the weak solution,we use the Hodge decomposition method to deal with X,the sub-space of H?curl;??,thus we can discuss the weak solutions in smaller space Y.And we can prove that the spaces Y and H?curl,??have the equivalent norms.Finally,according to the unique continuation theory,the Hodge decomposition method,and the compactly embedded lemma,we use the Lax-Milgram lemma and Fredholm alternative theorem to prove the existence and uniqueness of the weak so-lution of the equivalent variation problem by proving the continuity and coerciveness of the conjugate bilinear functional,ie there is the only weak solution of scattered electric field with the problem of the electromagnetic waves scattering in the bounded single cavity in Euclidean space R3.