A DECOMPOSITION OF L2(Ω)3 AND AN APPLICATION TO MAGNETOSTATIC EQUATIONS

Abstract

The domain Ω considered in the following is an open, bounded and connected subset of R3. The purpose of this paper is to find a decomposition of the space of functions L2(Ω)3 of the form grad P⊕ curl W and then to apply this result to the magnetostatic set of equations. Moreover, we prove that if the spaces P and W are chosen correctly, a function u of L2(Ω)3 can be written as grad p+curl w, p∈P and w∈W being unique (up to a constant for p). In the case of the magnetostatic equations, we provide a characterization of the magnetic field solution of these equations.