Boundary values and the transformation problem for constant principal strain mappings

We initiate a study of homeomorphisms f with constant principal strains (cps) between smoothly bounded planar domains D, D′. An initial result shows that in order for there to be such a mapping of a given Jordan domain D onto D′, a certain condition of an isoperimetric nature must be satisfied by the latter. Thereafter, we establish the fundamental fact that principal strain lines (characteristics) of such mappings necessarily have well-defined tangents where they meet ∂D. Using this, we obtain information about the boundary values of the Jacobian transformation of f, and finally we determine the class of all cps-homeomorphisms of a half-plane onto itself

© Hindawi Publishing Corp. BOUNDARY VALUES AND THE TRANSFORMATION PROBLEM FOR CONSTANT PRINCIPAL STRAIN MAPPINGS

We initiate a study of homeomorphisms f with constant principal strains (cps) between smoothly bounded planar domains D, D ′. An initial result shows that in order for there to be such a mapping of a given Jordan domain D onto D ′ , a certain condition of an isoperimetric nature must be satisfied by the latter. Thereafter, we establish the fundamental fact that principal strain lines (characteristics) of such mappings necessarily have well-defined tangents where they meet ∂D. Using this, we obtain information about the boundary values of the Jacobian transformation of f, and finally we determine the class of all cps-homeomorphisms of a half-plane onto itself

On Extremal Functions For John Constants

We define the John constant y(D) of a domain D cz C to be sup (a \u3e 1:1 \u3c /\u27(z)l ^a\u3e nD implies that/is univalent in D), and consider the case in which D is the upper half-plane H or a Jordan domain with sufficiently smooth boundary. A function /0is called an extremal function for such a D if 1 ^ /o(z)l ^ y(D) on D and f0(a)=f0(b) for two distinct points a, bedD. A simple compactness argument shows that there exist extremal functions for H. Let BN(D, a) = (/:/\u27 = e(XA, where Re(/i) is the harmonic measure of the union of N arcs on dD). It is shown that if/0is an extremal function for D, then/e BN(D, In y(D)) for some N. As a corollary we deduce that for any such D there exists K such that y(D) = sup(e“: All /in BK(D, a) are univalent in D); in particular this holds when D is the unit disk. © 1989, Oxford University Press. All rights reserved