Suppose โฆ is a Lipschitz domain in R n , n โฅ 2, g : โโฆ โ S n-1 is a Lipschitz map of degree d, and, for each p โ [n -1, n), u p : โฆ โ S n-1 is a p-energy minimizing map with u p | โโฆ = g. Then, for d = 0, u p necessarily has singularities, and
Here we discuss the behavior of u p and its singularities as p โ n and prove the following: THEOREM A For all p sufficiently close to n, depending on โฆ and g, the set sing u p of singularities (i.e., discontinuities) of u p contains precisely |d| points, all interior to โฆ. Also, for each a โ sing u p , deg[u p | โB r (a)] = (-1) sgn d for all sufficiently small positive r.
THEOREM B Any sequence of p's approaching n contains a subsequence p i so that the sets sing u p i converge to a set A of |d| points in โฆ, and the maps u p i converge strongly in C 1,ฮฑ on compact subsets of โฆ \ A for some positive ฮฑ < 1. The limit map u n belongs to C 1,ฮฑ (โฆ \ A, S n-1 ) โฉ C 0,ฮฑ (โฆ \ A) and is n-harmonic. THEOREM C For n โฅ 3, there are positive numbers N = N (โฆ, g) and ฮด = ฮด(โฆ, g) so that, for all p โ [n -1, n), any p-energy minimizer u p with u p | โโฆ = g has at most N singularities, and these are separated from each other and from โโฆ by a distance at least ฮด. (For n = 2 one must keep p bounded away from 1 [11, theorem D].) THEOREM D There is an asymptotic formula โฆ |โu p | p dx = |d| (n -1) p 2 ฯ n np + W g (a p 1 , . . . , a p |d| ) + o(np) as p โ n ,