Quasiconformal and quasiregular mappings
Non-linear potential theory
1) Based on the description of point multipliers given by Maz'ya and Shaposhnikova, we define analytic properties of a mapping F, inducing an isomorphism F* of spaces L of differentiable functions acting according to the change of variable rule F*f=f o F. As the space L, we considered the Sobolev space, the Besov space and the spaces of Bessel and Riesz potentials.
2) We obtained some results
related to the nonlinear potential theory associated with
solutions of the second order subelliptic equation
-div* A(x, Lu)=0. (1)
where Lu =(X1u,..., Xku) is a subgradient defined by smooth vector fields (X1u,..., Xku) satisfying the Hörmander hypoellipticity condition, and the mapping A satisfies some additional conditions.
Regularity of supersolutions of equation (1), as well as mutual relations between capacity in Sobolev spaces and the geometry of vector fields, are studied. In addition, metric and analytic conditions are obtained for removable singularities of bounded solutions of general equations, in particular, of the equations of the form (1). A-superharmonic functions possess the following basic properties: the comparison principle, the Harnack inequality for A-harmonic functions, the convergence theorems for monotone sequences, etc. These facts allow us to develop a theory that simila to the linear potential theory. Particular attention is focused on the questions connected with the geometry of vector fields.
3) Due to L. Ahlfors and L. Sario there is a classification of Riemannian surfaces depending on properties of the set of harmonic functions on a Riemannian surface. Some years ago I.Holopainen and S.Rickman have shown the way of generalization the above-mentioned classification to the case of Riemannian manifolds of arbitrary dimension. We used their idea to obtain the classification of subriemannian manifolds (M,g,D) based on characteristics of solutions of equation (1). Here M is an n-dimensional, non-compact, connected, orientable smooth manifold, g is a Riemannian tensor and D is a bracket generating k-dimensional tangent subbundle. We showed that there are inclusions between the subriemannian manifolds in accordance with the existence of a Green function to equation (1), the existence of non-constant positive (bounded) solutions to (1) or the presence of a non-constant solution with bounded Dirichlet's integral. The Carnot group is an example of subriemannian manifolds. As a consequence of the classification we deduce that nonhomeomorphic quasiconformal mappings of Carnot groups of different dimensions are constant.
4) We consider a bounded domain on a Carnot group with two disjoint compact sets in the closure of this domain. For this geometrical configuration we proved that the p-module of a family of curves connecting the compacts coincides with the p-capacity of this condenser. Generalizing concepts of the p-capacity and the p-module of a family of curves, we consider a p-module of family of vector measures on a Carnot groups (in Euclidean space firstly introduced by B. Fuglede, M. Ohtsuka, and H. Aikawa) and capacities associated with linear sub-elliptic equations. We proved continuity properties of p-module of vector measure, obtained some equalities and reciprocal relations between the above mentioned notions, that in particular case in Euclidean space reduce to classical relations, established by F. W. Gehring (p=n) and W. P. Zeimer (p<n, p>n).
5) We are interested also in studying quasiregular mappings (non-homeomorphic quasiconformal mapping) and their generalization on Carnot groups. Some estimates of the coefficient of distortion of quasiregular mapping, of a capacity of a condenser, and other characteristic were obtained. We showed that a quasiregular mapping with the coefficient of distortion close to 1 is a local homeomorphism under the condition that on the Carnot group any non-constant quasiregular mapping with the coefficient of distortion 1 is a global homeomorphism. We proved a lemma of type of Poletskii on an arbitrary Carnot group and gave some applications in the particular case of the Heisenberg group. The theory of quasiregular mappings (or in another terminology mapping with bounded distortion) in Euclidean space was founded and developed by Yu. G. Reshetnyak, O. Martio, S. Rickman.
6) We study the Hele-Shaw
problem for the plane dynamics of incompressible fluids
describing the two-dimensional flow in bounded or unbounded
simply connected plane domains. We obtained the differential
equation of Polubarinova-Galin type of free boundary of
above-mentioned moving flows for the case of non-zero surface
tension. The methods of the geometric function theory are
applied to prove that the property of the free boundary to be
starlike is preserved during the time of the existence
of solutions of the Polubarinova-Galin equation. We
constructed some new explicit solutions describing finite and
infinite bubbles in a two-dimensional corner flow.
7) We study the H-type
homogeneous groups related to the division algebras. The
parametric equations of sub-Riemannian geodesics constructed
making use of Hamiltonian method. The Lagrangian formalism for
quaternion and octonion H-type groups developed. The
sub-Riemannian invariants such like complex action function
and the volume element are found. The fundamental solutions to
the Laplace and Heat equations are presented. We also studied
the geodesics with additional constraint on velocity on the
unit sphere in 4-dimentional Euclidean space and
pseudo-hyperboloid in 4- dimensional space with pseudo-metric
with signature (--++). As a natural continuation of this work,
we introduced so called general H-type groups (pseudo H-type
groups), that are analogous of the H-type groups introduced by
A. Kaplan, but where instead of positive definite scalar
product we used any indefinite non-degenerate scalar product.
We showed that this tipe of groups admits an integer lattice,
that leads to the existence of new class of nilmanifolds,
obtaining by quotient of a general H-type group by the
8) My resent research
interests is related to the old mechanical problem of the
rolling of one body over another. For 3-dim body the problem
was studied by Cartan and then by several authors. The
multidimensional analogous describing rolling of bodies
embedded into Euclidean space can be found in the book of
Sharpe. We gave the intrinsic point of view on that problem
and described the distribution on the configuration space of
this mechanical problem, corresponding to the cinematic
constraints of no slipping an no twisting during the rolling.
We considered the rolling not only of Riemannian manifolds,
but also of semi-Riemannian manifolds.