We study the conformality problems associated with quasiregular mappings in space. Our approach is based on the concept of the infinitesimal space and some new Grötzsch-Teichmüller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients

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Generalized solvability of the classical boundary value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. Finally, some applications to the boundary value problems for A-harmonic functions are given.2010 MSC. 31A05, 31A20, 31A25, 31B25, 35Q15, 30E25, 31C05, 35F45.

Assume that Ω is a domain in the complex plane C and A(z) is symmetric 2×2 matrix function with measurable entries, det A = 1 and such that 1/K|ξ| 2 ≤ ⟨A(z)ξ, ξ⟩ ≤ K|ξ| 2 , ξ ∈ R 2 , 1 ≤ K < ∞. In particular, for semi-linear elliptic equations of the form div (A(z)∇u(z)) = f (u(z)) in Ω we prove Factorization Theorem that says that every weak solution u to the above equation can be expressed as the composition u = T • ω, where ω : Ω → G stands for a K−quasiconformal homeomorphism generated by the matrix function A(z) and T (w) is a weak solution of the semi-linear equation △T (w) = J(w)f (T (w)) in G. Here the weight J(w) is the Jacobian of the inverse mapping ω −1 . Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results in anisotropic media are given.