Functionals with Operator Curl in an Extended Magnetostatic Born--Infeld Model

Abstract

It is well-known that all the magnetostatic solutions with finite energy for the Born--Infeld model in $\Bbb R^3$ are trivial, and it was conjectured that incorporation of perturbation into the Born--Infeld functional could provide nontrivial solutions. In this paper we examine the extended Born--Infeld equations for the magnetostatic case in bounded domains. Under various boundary conditions the existence of nontrivial solutions is proved for small boundary data. The main feature of the extended Born--Infeld functionals is their degree one growth in the curl of the vector fields, which causes lack of weak compactness in the natural admissible spaces. To overcome this difficulty we introduce modified functionals and estimate their minimizers.