An N-dimensional version of the Beurling-Ahlfors extension

We extend monotone quasiconformal mappings from dimension n to n+1 while preserving both monotonicity and quasiconformality. The extension is given explicitly by an integral operator. In the case n=1 it yields a refinement of the Beurling-Ahlfors extension.Comment: 9 pages. Added references, edited concluding remark

Hopf differentials and smoothing Sobolev homeomorphisms

We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space $W^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.Comment: 22 pages, 0 figure

Harmonic mapping problem and affine capacity

The Harmonic Mapping Problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents considerable challenge and leads to several interesting open questions.Comment: 14 pages, 1 figur

Dynamics of quasiconformal fields

A uniqueness theorem is established for autonomous systems of ODEs, $\dot{x}=f(x)$, where $f$ is a Sobolev vector field with additional geometric structure, such as delta-monoticity or reduced quasiconformality. Specifically, through every non-critical point of $f$ there passes a unique integral curve.Comment: 26 pages, 1 figur

The Nitsche conjecture

The conjecture in question concerns the existence of a harmonic homeomorphism between circular annuli A(r,R) and A(r*,R*), and is motivated in part by the existence problem for doubly-connected minimal surfaces with prescribed boundary. In 1962 J.C.C. Nitsche observed that the image annulus cannot be too thin, but it can be arbitrarily thick (even a punctured disk). Then he conjectured that for such a mapping to exist we must have the following inequality, now known as the Nitsche bound: R*/r* is greater than or equal to (R/r+r/R)/2. In this paper we give an affirmative answer to his conjecture. As a corollary, we find that among all minimal graphs over given annulus the upper slab of catenoid has the greatest conformal modulus.Comment: 33 pages, 2 figures. Expanded introduction and references; added discussion of doubly-connected minimal surface

Harmonic mappings of an annulus, Nitsche conjecture and its generalizations

As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h \colon A(r,R) \to A(r_*, R_*)$ between planar annuli exists if and only if $\frac{R_*}{r_*} \ge {1/2} (\frac{R}{r} + \frac{r}{R})$. We prove this conjecture when the domain annulus is not too wide; explicitly, when $\log \frac{R}{r} \le {3/2}$. For general $A(r,R)$ the conjecture is proved under additional assumption that either $h$ or its normal derivative have vanishing average on the inner boundary circle. This is the case for the critical Nitsche mapping which yields equality in the above inequality. The Nitsche mapping represents so-called free evolution of circles of the annulus $A(r,R)$. It will be shown on the other hand that forced harmonic evolution results in greater ratio $\frac{R_*}{r_*}$. To this end, we introduce the underlying differential operators for the circular means of the forced evolution and use them to obtain sharp lower bounds of $\frac{R_*}{r_*}$.Comment: 26 page

Lipschitz regularity for inner-variational equations

We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the inner-stationary solutions need not be differentiable everywhere; the Lipschitz continuity is the best possible. But the proofs, even in the Dirichlet case, turn out to relay on topological arguments. The appeal to the inner-stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy-minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity; specifically, neo-Hookian type problems.Comment: No figure

On injectivity of quasiregular mappings

We give sufficient conditions for a planar quasiregular mapping to be injective in terms of the range of the differential matrix

Existence of energy-minimal diffeomorphisms between doubly connected domains

The paper establishes the existence of homeomorphisms between two planar domains that minimize the Dirichlet energy. Specifically, among all homeomorphisms f : R -> R* between bounded doubly connected domains such that Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an energy-minimal diffeomorphism. No boundary conditions are imposed on f. Although any energy-minimal diffeomorphism is harmonic, our results underline the major difference between the existence of harmonic diffeomorphisms and the existence of the energy-minimal diffeomorphisms. The existence of globally invertible energy-minimal mappings is of primary pursuit in the mathematical models of nonlinear elasticity and is also of interest in computer graphics.Comment: 34 pages, no figure

The role of recorded and verbal information in health information systems: A case study of the Expanded Program on Immunization in Mozambique

Background: There is ongoing interest in strengthening the informational component of the EPI as a mean to enhance the efficacy of service delivery. As developing country governments make significant investments in strengthening health information systems, benefits obtained from these initiatives tend to be below their potential. To improve this, it is argued that such systems need to take into account both formal (documented) and informal forms of information. Objective: This study measures the accuracy of people-based information with information recorded in the \"Road to Health Card\". Methods: 840 mothers were screened in a cross sectional study. Information about measles immunization status as documented in the Road to Health card was compared to mothers\' verbal information. The pattern of these answers among mothers presenting the card was compared with those among mothers without the card. Results: Measles vaccine information was 95.7% sensitive and 80.0% specific. Measles vaccination positive answer was 10.0% higher among mothers without the card. In the sample of mothers without the vaccination card, only 2.0% never received a card, 53.0% had forgotten to bring it, and the remaining 44.0% had lost it. Conclusion: A key implication of these findings is the need to develop a mother-specific information system that is based at the facility, and an integration of the verbal and tacit information sources while designing the information system. Further recommendation concerns the need to have more targeted information and education programs about the importance of the Road to Health Cards, especially focused for less educated mothers.The Ethiopian Journal of Health Development Vol. 21 (2) 2007: pp. 136-14