# Variational Theory for the Cavity Problem

# Variational Theory for the Cavity Problem

This chapter presents a standard variational method based on the electric field for the cavity problem to prepare for the finite element approximation of this problem. Mixed boundary conditions (perfectly conducting on one boundary, impedance on another) are assumed. A suitable solution space is described and the Helmholtz decomposition is used to decompose the problem into a simple scalar elliptic problem and a vector problem posed on a Sobolev space of divergence free fields. This space is shown to have a compact inclusion in the space of square integrable vector fields. After a proof of uniqueness of the solution, the Fredholm alternative is used to prove the existence of a solution to the variational problem, and hence show that this variational formulation is appropriate for discretization.

*Keywords:*
mixed boundary conditions, compact embedding, uniqueness, existence

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