Vol 210, No 3 (2019)

Along with a classical planar billiard, one can consider a topological billiard for which the motion takes place on a locally planar surface obtained by an isometric gluing of several planar domains along boundaries that are arcs of confocal quadrics. Here, a point is moving inside every domain along segments of straight lines, passing from one domain into another when it hits the boundary of the gluing. The author has previously obtained the Liouville classification of all such topological billiards obtained by gluings along convex boundaries. In the present paper, we classify all topological integrable billiards obtained by gluing both along convex and along nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics. For all such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules $W^*$) of Liouville equivalence are calculated. Bibliography: 25 titles.

The Dirichlet problem is considered in arbitrary domains for a class of second-order anisotropic elliptic equations with variable nonlinearity exponents and right-hand sides in $L_1$. It is proved that an entropy solution exists in anisotropic Sobolev spaces with variable exponent. It is proved that the entropy solution obtained is a renormalized solution of the problem under consideration. Bibliography: 37 titles.