Slit Maps in the Study of Equal-Strength Cavities in $n$-Connected Elastic Planar Domains

Abstract

The inverse problem of plane elasticity on $n$ equal-strength cavities in a plane subjected to constant loading at infinity and in the cavities' boundaries is analyzed. By reducing the governing boundary value problem to the Riemann--Hilbert problem on a symmetric Riemann surface of genus $n-1$, a family of conformal mappings from a parametric slit domain onto the $n$-connected elastic domain is constructed. The conformal mappings are presented in terms of hyperelliptic integrals, and the zeros of the first derivative of the mappings are analyzed. It is shown that for any $n\ge 1$ there always exists a set of the loading parameters for which these zeros generate inadmissible poles of the solution.