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

Abstract

Abstract. We will prove a global estimate for the gradient of the solution to the Poisson differential inequality |∆u(x)| ≤ a|∇u(x)| 2 + b, x ∈ B n , where a, b < ∞ and u| S n−1 ∈ C 1,α (S n−1 , R m ). If m = 1 and a ≤ (n + 1)/(|u|∞4n √ n), then |∇u| is a priori bounded. This generalizes some similar results due to E. Heinz ([13]) and Bernstein ([3]) for the plane. An application of these results yields the theorem, which is the main result of the paper: A quasiconformal mapping of the unit ball onto a domain with C 2 smooth boundary, satisfying the Poisson differential inequality, is Lipschitz continuous. This extends some results of the author, Mateljević and Pavlović from the complex plane to the space.

On Pavlovic’s Theorem in Space

“…In higher dimensions Pavlovic's approach seems difficult to work with; instead it would seem conceivable that the Lipschitzproperty follows by the regularity theory of elliptic PDE's. In fact, such an approach was done by Kalaj [15]. However, the proof in [15] is rather long and technical, and one of the purposes of this note is to give a simple and self-contained argument showing the Lipschitz property in all dimensions.…”

“…The paper has initiated an extensive investigation between the Lipschitz conditions and harmonic quasiconformal mappings, see e.g. [3], [6], [14], [15], [17], [22] and their references.…”

On a Quasi‐isometry for Quasiconformal Mapping Satisfying Poisson Differential Inequality

“…Remark It is noted that both harmonic mappings and twice continuously differentiable mapping f satisfying Poisson's equation are contained in the class of functions satisfying inequality (5). We refer the reader to see the research works on those mappings (5) for two dimensions [3, 8] and higher dimensions in [11]. …”

Quasiconformal maps with controlled Laplacian

“…Conditions (15) follows easily by employing the formulas for the derivatives of the implicit function, after first approximating by smooth functions. Note, in regards to condition (16), we note that in general the inverse of a harmonic diffeomorphism needs not to be harmonic, so (16) is not a direct consequence of (14). However, the first condition in (13) tells us that the maximal complex dilatation k n of Ψ n tends to 0 as n → ∞, so that Ψ n is asymptotically conformal and this makes (16) more plausible.…”

“…Together, our Theorems 1 and 2 considerably improve the main result of the first author and Pavlović from [15], where it was instead assumed that ∆w ∈ C(D). Other related results are contained in [14], we refer to [6] and references therein for other type of connections between quasiconformal and Lipschitz maps. In order to state our last theorem, we recall the result of V. I. Smirnov, stating that a conformal mapping of the unit disk U onto a Jordan domain Ω with rectifiable boundary has a absolutely continuous extension to the boundary.…”