On Certain Nonlinear Elliptic PDE and Quasiconformal Maps Between Euclidean Surfaces

Abstract

We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper (Kalaj and Mateljević, J Anal Math 100:117-132, 2006) have important role in this paper.

A priori estimate of gradient of a solution of a certain differential inequality and quasiconformal mappings

“…and satisfies Pisson differential inequality, then the function χ(x) = −d(u(x)), where d(u) = dist(u, ∂Ω), satisfies as well Pisson differential inequality in some neighborhood of the boundary. By using this fact and Theorem B we prove Theorem C. This extends some results of the author, Mateljevic and Pavlovic ([23], [26], [20], [21] and [32]) from the plane to the space. It is important to notice that, the conformal mappings and decomposition of planar harmonic mappings as the sum of an analytic and an anti-analytic function played important role in establishing some regularity boundary behaviors of q.c.…”

“…One of the starting points of this paper is the following theorem which was one of the main tools in proving some recent results of the author and Mateljevic (see [21] and [20] [3] and [13]). Let s : U → R (s : U → B m ) be a continuous function from the closed unit disc U into the real line (closed unit ball) satisfying the conditions:…”

On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

“…O. Martio [17] was the first who considered harmonic quasiconformal mappings on the complex plane. Recent papers [10], [12], [14], [21] and [13] bring much light on the topic of quasiconformal harmonic mappings on the plane. See also [11] for the extension of the problem on the space.…”

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

“…(5) If w is a quasi-conformal mapping between two C 2,α Jordan domains satisfying the partial differential inequality |∆w| B|∇w| 2 + Γ , then w is Lipschitz (a result proved in [19]). …”