Logarithmic asymptotics of a class of mappings
The author studies the asymptotic behavior of lower $Q$-homeomorphisms relative to a $p$-modulus in $\mathbb R^n$ and its applications. Denote by $N(S,y)$, $y\in\mathbb R^n$, the multiplicity of the covering of $y$ by a $k$-dimensional surface $S$ in $\mathbb R^n$. Then $\int_S\rho\, d\mathcal A_k:=\int_{\mathbb R^n}\rho(y)N(S,y)dH^ky$ with a Borel function $\rho:\mathbb R^n\to[0,\infty)$ and the Hausdorff measure $H^k$. Let $p>1$. The $p$-modulus of a family $\Gamma$ of $k$-dimensional surfaces $S$ is the value $M_p(\Gamma)=\inf\int_{\mathbb R^n}\rho^p(x)dm(x)$, where the infimum is taken over all $\rho$ such that $\int_S\rho^k\,d\mathcal A_k\geq1$ for every $S\in\Gamma$. A property $P$ holds for $p$-almost every $k$-dimensional surface $S\in\Gamma$ if $P$ can fail for a subfamily $\Gamma'\subset\Gamma$ only when $M_p(\Gamma')=0$. Let $D$ and $D'$ be domains in $\mathbb R^n$, $n\geq2$, $0\in D$, and let $Q:D\to(0,\infty)$ be measurable. A homeomorphism $f:D\to D'$ is called a lower $Q$-homeomorphism relative to the $p$-modulus if $M_p(f\Sigma_{\mathbb A})\geq\inf\int_{\mathbb A}\rho^p(x)/Q(x)\,dm(x)$ for every $\mathbb A=\mathbb A(\epsilon_1,\epsilon_2)=\{x\in\mathbb R^n:\epsilon_1<|x|<\epsilon_2\}$, $0<\epsilon_1\leq\epsilon_2<d_0=\text{dist}(0,\partial D)$, $\Sigma_{\mathbb A}$ is the family of all spheres $S(r)=\{x\in\mathbb R^n:|x|=r\}$, $r\in(\epsilon_1,\epsilon_2)$, and the infimum is taken over all measurable $\rho:\mathbb R^n\to[0,\infty)$ such that $\int_S\rho^{n-1}(x)d\mathcal A_{n-1}\geq1$ for $p$-almost every $(n-1)$-dimensional surface $S\in\Gamma$. Set \[ \|Q\|_{(n-1)/(p-n+1)}(r)= \left(\int_{S_r}Q^{(n-1)/(p-n+1)}(x)d\mathcal A_{n-1}\right)^{(p-n+1)/(n-1)}. \] The main result gives lower limit estimates in the following theorem. \par Theorem 3.1. Let $n\geq2$ and $p>n$. Assume that $Q:\mathbb B^n\to(0,\infty)$ is a measurable function on the unit ball $\mathbb B^n$ and $\|Q\|_{(n-1)/(p-n+1)}(r)<\infty$ for a.e. $r\in(0,\epsilon_0)$, $\epsilon_0\in(0,1)$. Let $f:\mathbb B^n\to\mathbb B^n$ be a lower $Q$-homeomorphism relative to the $p$-modulus, $f(0)=0$. If \[ \int_{\mathbb A(\epsilon_1,\epsilon_2)}\frac{Q^{(n-1)/(p-n+1)}(x)dm(x)}{|x|^{p/(p-n+1)}}\leq C_0\log^{\kappa}(\epsilon_2/\epsilon_1) \] holds for some $\kappa\in[0,p/(p-n+1)]$, $C_0\in(0,\infty)$ and any $0<\epsilon_1<\epsilon_2<\epsilon_0$, then \[ \liminf_{x\to0}|f(x)|\log^{\theta}(1/|x|)\leq\nu_0C_0^{\gamma}, \] where $\gamma=\frac{p-n+1}{(n-1)(p-n)}$, $\theta=\frac{p-\kappa(p-n+1)}{(n-1)(p-n)}$, and $\nu_0>0$ depends only on $n$, $p$ and $\kappa$. \par Theorem 3.1 is applied to estimates in the Orlicz-Sobolev classes.
\(p\)-modulus of families of \(k\)-dimensional surfaces
scientific article; zbMATH DE number 7000124