Boundary property of $n$-dimensional mappings with bounded distortion

V. M. Miklyukov

Matematičeskie zametki, Tome 11 (1972) no. 2, pp. 159-164

Résumé

The following assertion is proved: let $f: B\to R^n$ be an arbitrary (in general, not single-sheeted) mapping with bounded distortion of an $n$-dimensional sphere $B$, satisfying the conditions: A) the set $f(B)$ is bounded; B) the partial derivatives $\frac{\partial f_i}{\partial x_j}$ ($i,j=1,2,\dots,n$) are summable with respect to $B$ with degree $\alpha$ ($1<\alpha\leqslant n$). Then the mapping $f$ has angular boundary values everywhere on the boundary of the sphere with the possible exception of a set of $\alpha$-capacity zero.