Variational problems with multiple-valued functions and mappings
Hardt, Robert M.
Doctor of Philosophy thesis
The thesis discusses the regularity problems of Dirichlet-minimizing multiple-valued mappings and stationary-harmonic multiple-valued functions. It consists of two chapters: Chapter 1. Interior partial regularity of Dirichlet-minimizing multiple-valued maps from the ball to the sphere. For m, n ≥ 2 and assuming Sn⊂Rn +1 , we show that any m-dimensional Dirichlet-minimizing QQ( Sn )-valued map is Holder continuous in the interior of the domain except for a closed subset S whose Hausdorff dimension is less than m - 2, i.e. Hm-2 (S) = 0. Furthermore, the Hausdorff dimension of the branch set B of any Dirichlet-minimizing Q Q( Sn )-valued map is no greater than m - 2 in the domain O, i.e. Hm-2+e (B) = 0, ∀ epsilon > 0. For m = 2, the branch set B is locally finite in the interior of the domain O, i.e. H0 (B) < infinity. Chapter 2. Interior regularity of 2-dimensional stationary harmonic multiple valued functions. Any two-dimensional stationary harmonic multiple-valued function f is Holder continuous in the interior of the domain. In the 1-dimensional case, the set of branch points of any 1-dimensional stationary-harmonic multiple-valued function f is at most locally finite and f is a graph which is locally a finite union of straight line segments.