On quasi-conformal self-mappings of the unit disc and elliptic PDEs in the plane (Q2866550)

It is well known that \(K\)-quasi-conformal mappings between smooth domains are Hölder continuous and the Hölder constant is \(\frac{1}{K}\). What additional condition can ensure a quasi-conformal mapping to be Lipschitz continuous? Harmonicity of the mapping or a partial differential inequality are conditions to be assumed. There have been some related results, such as, a harmonic quasi-conformal mapping of the unit disc onto itself is Lipschitz. This paper gives a condition for a quasi-conformal mapping of the unit disc onto itself to be Lipschitz in terms of the elliptic partial differential inequality \(|L[\omega]|\leq \mathcal {B}|\nabla\omega|^2+\Gamma\). This is an extension of the class of harmonic quasi-conformal mappings, where, in the inequality, \(L\) is the Laplace operator and \(L[\omega]=0\).NEWLINENEWLINEFor the proof of the main theorem, one has to show an a priori bound for the gradient of the solution of the PDE. The authors first show that the gradient has an a priori bound on the compact sets of the unit disc in terms of constants of the elliptic PDE and the modulus of continuity of the solution function. By Mori's theorem, the modulus of continuity of a \(K\)-quasi-conformal mapping of the unit disc onto itself depends only on \(K\). So they get an a priori bound depending on constants of the PDE and \(K\) on the compact sets of the unit disc. To show the a priori bound of the gradient in some neighborhood of the boundary of the unit disc the authors use a theorem of Nagumo, recalled as Theorem 2.8 in Section 2.