Estimates for Jacobian and dilatation coefficients of open discrete mappings with controlled \(p\)-module
Given a family \(\Gamma\) of paths in \(\mathbb{R}^n\) and some \(p>1\), the \(p\)-modulus is the quantity \[ \mathcal{M}_p(\Gamma) = \inf\left\{ \int_{\mathbb{R}^n}\rho^p(x) \, dm(x)\right\} \] where the infimum is taken over all Borel functions \(\rho:\mathbb{R}^n \to [0,\infty]\) such that \(\int_{\gamma}\rho \geq 1\) for all \(\gamma \in \Gamma\). Such functions \(\rho\) are called admissible functions for \(\Gamma\). In the paper under review, the authors consider two classes of maps which, although weaker than quasiregular maps, still retain many nice analytic properties that quasiregular maps satisfy. Fix a domain \(D\subset \mathbb{R}^n\), a Lebesgue-measurable map \(Q:D \to [0,\infty]\) and some \(p\geq 1\). A discrete open map \(f:D\to\overline{\mathbb{R}^n}\) is a \((p,Q)\)-map if for all path families \(\Gamma\) in \(D\) and all admissible \(\rho\) for \(\Gamma\), \[ \mathcal{M}_p(f(\Gamma)) \leq \int_D Q(x) \rho^p(x) \, dm(x). \] Slightly weaker, a discrete open map \(f:D\to\overline{\mathbb{R}^n}\) is a ring \((p,Q)\)-map if for all annuli \(A=A(x_0,r,R)\subset D\) and for all path families \(\Gamma\) with curves in \(A\) joining the two components of \(\partial A\), there exists an admissible \(\rho(x) = \eta(|x-x_0|)\) for \(\Gamma\) such that \[ \mathcal{M}_p(f(\Gamma)) \leq \int_A Q(x)\eta^p(|x-x_0|) \, dm(x). \] It is known that both classes are ACL and therefore differentiable almost everywhere. For those \(x\in D\) that \(f'(x)\) is defined, \(\ell(f'(x))\) (resp. \(\|f'(x)\|\)) denotes the smallest (resp. biggest) modulus of the complex eigenvalues of \(f'(x)\) while \(J(x,f)\) denotes the Jacobian of \(f\) at \(x\). For the \((p,Q)\)-maps with \(Q\in L^1_{\text{loc}}(D)\), the authors prove in Theorems 1.2 and 4.1 that if \(1 < p < n\), then \[ C^{-1} Q(x)^{\frac{n}{n-p}} \leq |J(x,f)| \leq C Q(x)^{\frac{n}{n-p}} \quad\text{ a.e. in }D \] for some \(C=C(n,p)\) while if \(n-1 < p\leq n\), then \[ 0 \neq |J(x,f)| \leq Q(x) \ell(f'(x))^{p} \quad\text{ a.e. in }D. \] For the ring \((p,Q)\)-maps with \(Q\in L^1_{\text{loc}}(D)\), the authors prove (Theorem 1.1) that when \(n-1<p<n\), then \[ \max\{\|f'(x)\|^{n}, |J(x,f)|\} \leq C Q^{\frac{1}{n-p}}\quad\text{ a.e. in }D \] for some \(C=C(n,p)\). The authors also give examples that illustrate the sharpness of restrictions on \(p\) for Theorem 1.1.
scientific article; zbMATH DE number 6814624
scientific article