Existence and regularity of solutions of d ! = f with Dirichlet boundary conditions

Given a bounded open set Ω ⊂ ℝ n and a (k + l)-form f satisfying some compatibility conditions, we solve the problem (in Holder spaces) $$ d\omega = f\;in\;\Omega, \quad \omega = 0\;on\;\partial \Omega $$ .

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