Let \(\Omega\subset{\mathbb{R}}^2\) be a bounded Jordan domain, and let \({\mathcal{Q}}\subset {\mathbb{R}}^2\) be a convex domain. Let \(h=u+iv: \partial \Omega\to \partial {\mathcal{Q}}\) be a homeomorphism of the boundaries. The Radó-Kneser-Choquet theorem says that the continuous harmonic extension \(H: \Omega\to {\mathcal{Q}}\) of \(h\) is a \(\text{{C}}^\infty\) diffeomorphism. There are extensions of this result for linear and nonlinear elliptic PDEs assuming \(\Omega\) is a Jordan domain. The authors are interested in the case where \(\Omega\) is only simply connected. Since in this case one cannot speak of boundary homeomorphisms, the authors consider monotone boundary maps instead. Let \(h: \partial\Omega\to \partial{\mathcal{Q}}\) be a continuous monotone mapping, where \(\Omega\subset {\mathbb{R}}^2\) is a simply connected domain and \({\mathcal{Q}}\subset {\mathcal{R}}^2\) is a bounded convex domain. The main result of this paper says that the \(p\)-harmonic extension \(H: \Omega\to{\mathbb{R}}^2\), \(1< p<\infty\), is a \(\text{{C}}^\infty\) diffeomorphism onto \({\mathcal{Q}}\).