Lipschitz maps with prescribed local Lipschitz constants

Let \(\Gamma\) be a closed subset of a complete Riemannian manifold \(M\) of dimension \(\geq 2\), let \(f: M \to N\) be a Lipschitz map to a complete Riemannian manifold \(N\), and let \(\psi\) be a continuous function which dominates the local Lipschitz constant of \(f\). We construct a Lipschitz map which agress with \(f\) on \(\Gamma\) and whose local Lipschitz constant is \(\psi\).