Continuity in the sense of Kudryavtsev for elliptic mappings (Q5930781)

A mapping \(f:G \rightarrow E^n\;(G\) a domain of the Euclidean \(n\)-space \(E^n)\) of the Sobolev class \(W^1_n(G)\) is called elliptic [cf. \textit{A. P. Oskolkov}, Tr. Mat. Inst. Steklov 92, 182-191 (1966; Zbl 0156.11702)] if for almost all \(x\in G\) we have its Jacobian \(Jf(x)>0\) and \[ |\nabla f(x)|^n \leq (nQ)^{{n\over 2}}Jf(x)+Q_1(x)|f(x)|^n+Q_2(x), \] where \(|\nabla f(x)|\) is the Euclidean norm of the Jacobi matrix of \(f\) at \(x\), \(Q\geq 1\), \(Q_1(x), Q_2(x)\geq 0\) and \(Q_1, Q_2\in L_q(G), q>1.\) A continuous map \(f:G \rightarrow E^n\) is said to be continuous in the sense of Kudryavtsev with index \(\gamma\in (0,1)\) if the inequality \[ \varlimsup_{r\rightarrow 0} {{\text{osc}_{n-1}f(x,r)}\over{r^{\gamma (n-1)}}}\leq \text{Const} < \infty , \] holds for all \(x\in G,\) where \(\text{osc}_{n-1}f(x,r) = \inf _{e\in\Omega} \text{Vol}_{n-1}\{ f[S_{n-2}(x,r,e)]\} ,\) is the \((n-1)\)-oscillation of \(f,\) \(\Omega\) is the set of all unit vectors \(e = {x\over{|x|}}\) and \(S_{n-2}(x,r,e)\) is the diametrical cross-section of the sphere \(S_{n-1}(x,r)\) of centre \(x\) and radius \(r\) by the hyperplane that passes through the point \(x\) and whose normal is \(e.\) Kudryavtsev's \(p\)-variation \((p\geq 1)\) of a homeomorphism \(f:G\rightarrow E^n\) is \(V_G^p= \int_G|Jf(x)|^p dx\) [cf. \textit{L. D. Kudryavtsev}, Usp. Mat. Nauk 10, No. 2(64), 167-174 (1955; Zbl 0064.29401)]. The author establishes the following theorem: ``If a bounded homeomorphic elliptic mapping \(f:G \rightarrow E^n\) has a bounded Kudryavtsev \(p\)-variation with \(p = {{\alpha (n-1)}\over{n(\alpha -1)}}\), \(1 < \alpha <2,\) then \(f\) is continuous in the sense of Kudryavtsev with index \(\gamma = {{n-\alpha}\over{\alpha (n-1)}}\)''.