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 domainD ontoD′, 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.