A Global Version Of The Darboux Theorem With Optimal Regularity And Dirichlet Condition

Let n > 2 be even; r >= 1 be an integer; 0 < alpha < 1; Omega be a bounded, connected, smooth, open set in R-n; and nu be its exterior unit normal. Let f, g is an element of C-r,C-alpha((Omega) over bar; Lambda(2)) be two symplectic forms (i.e., closed and of rank n) such that f-g is orthogonal to the harmonic fields with vanishing tangential part, nu boolean AND f,nu boolean AND g is an element of C-r+1,C-alpha(partial derivative Omega; Lambda(3)) and nu boolean AND f = v boolean AND g on partial derivative Omega. Moreover assume that tg + (1-t)f has rank n for every t is an element of [0, 1]. We will then prove the existence of a phi is an element of Diff(r+1,alpha)((Omega) over bar; (Omega) over bar )satisfying