A remark on H¹ mappings

With {Mathematical expression}, we here construct, for each positive integer N, a smooth function {Mathematical expression} of degree zero so that there must be at least N singular points for any map that minimizes the energy {Mathematical expression} in the family {Mathematical expression}. The infimum of ε over U(g) is strictly smaller than the infimum of ε over the continuous functions in U(g). There are some generalizations to higher dimensions.