Div-Curl System with Potential and Maxwell-Stokes System with Natural Boundary Condition

摘要

This paper concerns the boundary value problems of div-curl system with potential, and those of the Maxwell-Stokes system under natural boundary condition. One of the main features of these systems is that the equations contain an unknown potential term, which represents the effect of domain topology. Solvability of these systems depends not only on the structure of the equations and the type of the boundary conditions, but also on the domain topology. We show existence of solutions of these systems by using variational methods together with a modified de Rham lemma, and by methods of compact operators. Regularity of the solutions and the eigenvalue problem are also discussed.

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