Variational Problems And Partial Differential Systems Involving Operator Curl

The partial differential systems involving operator curl have extensive applica-tions in mathematical physics. For example, Maxwell’s equation in classical elec-trodynamics, Born-Infeld model in nonlinear electrodynamics and several models describing the qualities of superconductors. On the other hand, from the view of mathematics, the problems involving operator curl are also well worth explored. We observe that they are quite different with problems involving operator gradient. Besides, they have delicate relation with other fields, such as Lame operator, and the corresponding regularity theory is closely related to the basic theory of general elliptic equations (e.g. p-Laplace equation and p-growth equation with Dirichlet boundary condition, Neumann boundary condition, and also the oblique derivative problem).On account of the above reasons, we try to study the relation between operator curl, topology of the domain, lower order term and the boundary conditions, and also how they impact on the existence and regularity of solutions. So we can further understand the common points and differences between the problem involving op-erator curl and operator gradient, establish the basic theory of variational problems and partial differential equations involving operator curl, and also provide new ideas of related physical problems.The thesis is divided into two parts. In the first part we study an extended Born-Infeld model in bounded domains, which has clear physical background and plays an important role in nonlinear electrodynamics. We overcome several difficulties, such as order-one growth of the leading term and lack of weakly compactness of the functional, and degenerate ellipticity of the corresponding system, and obtain non-trivial classical solutions with finite energy. In the second part we study q-curl curl quasilinear systems, which is not only of typical structure from mathematical view-point, but also related to Bean’s critical state model for hard superconductors. We study the existence and regularity of the solutions under two kinds of boundary conditions, and observe its relationship with several classical elliptic problems, over-come the singularity (1<q<2) or degeneracy (2≤q<+∞) of the leading term of the system, and establish systematic mathematical theory.