Minimisers and Kellogg's theorem (Q785360)

Let \(0<\alpha <1\) and \(n\in \mathbb{N}\). The Kellogg-Warschawski theorem says that, if \(g\) is a conformal map between Jordan domains with \(C^{n,\alpha }\) boundaries, then \(g\) is \(C^{n,\alpha }\) up to the boundary. The authors establish the following analogues of this result for minimizers of Dirichlet energy \(\mathcal{E}[f]=2\int_{D}(\left\vert \partial f\right\vert ^{2}+\left\vert \overline{\partial }f\right\vert ^{2})~dA\) among diffeomorphisms \(f:D\rightarrow \Omega \) between doubly connected plane domains. Let \(\mathrm{Mod}(D)\) denote the conformal modulus of \(D\). Theorem 1.1. If \(D,\Omega \) have \(C^{1,\alpha }\) boundaries and \(f\) is a diffeomorphic minimizer of \(\mathcal{E}[f]\) among all diffeomorphisms from \(D\) to \(\Omega \), then \(f\) has a \(C^{1,\alpha ^{\prime }}\) extension up to the boundary, where \(\alpha ^{\prime }=\alpha \) if \(\mathrm{Mod}(D)\geq \mathrm{Mod}(\Omega )\) and \(\alpha ^{\prime }=\alpha /(2+\alpha )\) if \(\mathrm{Mod}(D)<\mathrm{Mod}(\Omega )\). Theorem 1.2. If \(D,\Omega \) have \(C^{n,\alpha }\) boundaries, \(\mathrm{Mod}(D)\geq \mathrm{Mod}(\Omega )\) and \(f\) is a diffeomorphic minimizer of \(\mathcal{E}[f]\) among all diffeomorphisms from \(D\) to \(\Omega \), then \(f\) has a \(C^{n,\alpha }\) extension up to the boundary.