On two-dimensional ferromagnetism

We study the existence of minimizers for 2-dimensional ferromagnetism with various assumptions on different terms of the total energy. Our general philosophy for finding the minimizing magnetizations is to reduce this problem to a minimization problem for a new energy over a divergence-free field. Such a general philosophy works for all dimensions. However, in the twodimensional case, since the divergence-free fields are equivalent to the rotated gradients, this new energy can be written as an integral functional of gradients and hence the minimization problem can be solved by some non-convex minimization procedures in the calculus of variations. We will apply this general philosophy to the two-dimensional case in this paper and to the three-dimensional situation in a future work. Special emphasis is placed on the analysis of existence/non-existence depending on the applied field H and the physical domain Ω.