A quantitative estimate for mappings of bounded inner distortion

Abstract

We establish a quantitative version of distortion inequality for mappings of bounded inner distortion. Some applications to the integral form of the isoperimetric inequality are given.

Appendix

We prove (24). We recall the following algebraic inequality

$$\begin{aligned} 2 \left\langle |\xi |^{p-2}\xi - |\zeta |^{p-2}\zeta , \xi - \zeta \right\rangle \ge \left( |\xi |^{p-2}+|\zeta |^{p-2} \right) |\xi -\zeta |^2, \end{aligned}$$

(93)

for all $\xi ,\zeta \in {\mathbb {R}}^n$. Consider the positive square root of $A(x)$, a symmetric positive definite matrix $\sqrt{A(x)}$ such that $\sqrt{A(x)} \cdot \sqrt{A(x)} = A(x)$ for a.e $x \in \varOmega$. Thus

$$\begin{aligned} \left\langle A(x)\xi ,\xi \right\rangle = \left\langle \sqrt{A(x)}\xi ,\sqrt{A(x)}\xi \right\rangle = \left| \sqrt{A(x)}\xi \right| ^2 \end{aligned}$$

Now,

$$\begin{aligned}&\left\langle {\mathcal { H}} (x,\xi ) - {\mathcal { H}} (x,\zeta ),\xi -\zeta \right\rangle \\&\quad = \left\langle \left| \sqrt{A(x)} \xi \right| ^{p-2} \sqrt{A(x)} \xi - \left| \sqrt{A(x)} \zeta \right| ^{p-2} \sqrt{A(x)} \zeta , \sqrt{A(x)} \xi - \sqrt{A(x)} \zeta \right\rangle \\&\quad \ge \frac{1}{2} \left( \left| \sqrt{A(x)} \xi \right| ^{p-2} + \left| \sqrt{A(x)} \zeta \right| ^{p-2} \right) \left| \sqrt{A(x)} \xi -\sqrt{A(x)} \zeta \right| ^2 \\&\quad \ge \frac{a^p}{2} |\xi -\zeta |^2 \left( |\xi |^{p-2}+|\zeta |^{p-2} \right) \end{aligned}$$

The proof of (24) follows similarly, by using the algebraic inequality

$$\begin{aligned} \left| |\xi |^{p-2} \xi - |\zeta |^{p-2} \zeta \right| \le (p-1) |\xi -\zeta | \left( |\xi |+|\zeta |\right) ^{p-2}, \end{aligned}$$

(94)

for all $\xi ,\zeta \in {\mathbb {R}}^n$.