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According to the Eshelby conjecture, an ellipse or ellipsoid is the onlyshape that induces an interior uniform strain under a uniform far-fieldloading. We extend the Eshelby conjecture to domains of general shape foranti-plane elasticity. Specifically, we show that for each positive integer$N$, an inclusion induces an interior uniform strain under a polynomial loadingof degree $N$ if and only if the exterior conformal map of the inclusion is aLaurent series of degree $N$. Furthermore, for the isotropic case, wecharacterize the shape of an inclusion by only using the first-degreepolynomial loading and explicitly solve the interior potential of the inclusionin terms of the Grunsky coefficients.

An extension of the Eshelby conjecture to domains of general shape in anti-plane elasticity