On one modulus inequality for mappings with finite length distortion

The V?is?l? inequality, which is well known in the theory of quasilinear mappings, is extended to the class of mappings with finite length distortion.     

Compactness theory and mappings with finite length distortion

The present paper is devoted to the study of mappings with finite length distortion introduced in 2004 by O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov. It is proved that the locally uniform limit of homeomorphisms with finite length distortion is a homeomorphism or a constant provided that the so-called inner dilatations of the sequence of homeomorphisms are almost everywhere (a.e.) majorized by a locally integrable function. In particular, it is studied the pointwise behavior of the so-called outer dilatations. For these dilatations, the pointwise semicontinuity and semicontinuty in the mean are proved. It is also proved some theorems on the convergence of matrix dilatations.     

Mappings of finite distortion: The Rickman-Picard theorem for mappings of finite lower order

We show that an entire mappingf of finite distortion with finite lower order can omit at most finitely many points when the distortion function off is suitably controlled. The proof uses the recently established modulus inequalities for mappings of finite distortion [15] and comparison inequalities for the averages of the counting function. A similar technique also gives growth estimates for mappings having asymptotic values.     

On the theory of mappings with finite area distortion

It is shown that mappings in ℝⁿ with finite distortion of area in all dimensions 1 ≤ k ≤ n − 1 satisfy certain modulus inequalities in terms of inner and outer dilatations of the mappings; in particular, generalizations of the well-known Poletskii inequality for quasiregular mappings are proved. The theory developed is applicable, for example, to the class of finitely bi-Lipschitz mappings, which is a natural generalization of the bi-Lipschitz mappings, as well as isometries and quasi-isometries in ℝⁿ.     

On the weak limit of mappings with finite distortion

We give a new proof that the limit of a weakly convergent sequence of mappings with finite distortion also has finite distortion. The result has been recently proved by Gehring and Iwaniec using the biting convergence of Jacobians. We present a different proof using simply the lower semi-continuity of quasiconvex functionals.

     

Slow mappings of finite distortion

We examine mappings of finite distortion from Euclidean spaces into Riemannian manifolds. We use integral type isoperimetric inequalities to obtain Liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds.     

Mappings with finite length distortion

General properties of mappings of finite metric distortion and of finite length distortion are studied. Uniqueness, equicontinuity, boundary behavior and removability of singularities are obtained under minimal additional assumptions.     

On the convergence of mappings with k-finite distortion

Mathematical Notes -     

Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion

We study the problem of minimizing the functional \(I(\phi ) = \int\limits_\Omega {W(x,D\phi )dx}\) on a new class of mappings. We relax summability conditions for admissible deformations to φ ∈ W _n ¹ (Ω) and growth conditions on the integrand W(x, F). To compensate for that, we require the condition \(\frac{{\left| {D\phi (x)} \right|^n }} {{J(x,\phi )}} \leqslant M(x) \in L_s (\Omega )\), s > n ? 1, on the characteristic of distortion. On assuming that the integrand W(x, F) is polyconvex and coercive, we obtain an existence theorem for the problem of minimizing the functional I(φ) on a new family of admissible deformations A.     

On a certain differential property of mappings with bounded distortion

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 6, pp. 199–200, November–December, 1992.     

Mappings of finite distortion: Removable singularities for locally homeomorphic mappings

Let be a locally homeomorphic mapping of finite distortion in dimension larger than two. We show that when the distortion of satisfies a certain subexponential integrability condition, small sets are removable. The smallness is measured by a weighted modulus.

     

Bloch’s theorem for mappings of bounded and finite distortion

We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural form. Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author was visiting at the University of Michigan. He wishes to thank the department for hospitality.     

Positive results on discreteness and openness for mappings of finite distortion

In this paper, we prove some positive results on discreteness and openness of mappings of finite distortion under some integrability condition on the distortion and the multiplicity function of the mapping. We also show that in some sense, our results are sharp.     

On deviations and defects of meromorphic functions of finite lower order

We obtain estimates for the sum of deviations and sum of defects to power 1/2 in terms of the Valiron defect of the derivative at zero. In particular, the Fuchs hypothesis (1958) is verified.