= Hilbert problem for an antiplane crack
{c}
{title2=$G^++G^-=-2p$}
For a crack on the real axis under applied shear traction $p$, write $G=\mu F'$ for the additional <complex stress potential for antiplane shear> derivative. Reflection of the induced displacement gives $G^- =\overline{G^+}$. Traction-free crack faces therefore require $G^++G^-=-2p$; intact portions have $G^+=G^-$. This is a scalar <Riemann-Hilbert problem>. Endpoint regularity, single-valued displacement and the absence of additional remote loads fix its homogeneous terms.
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