Title: Complete Solutions to Conjectures of Polya-Szeg"o and Eshelby in Two Dimensions

Abstract : Eshelby conjectured in 1961 that if for a given uniform loading the field inside an elastic inclusion is uniform, then the inclusion must be an ellipse or an ellipsoid. On the other hand, P\`olya and Szeg\"o conjectured in 1951 that if the polarization tensor associated with an inclusion has the minimal trace, then the inclusion must be a disk or a ball. We prove both conjectures in two dimensions. We show that if the polarization tensor has the minimal trace, then the field inside the inclusion must be uniform. We then show that if the (elastic or electric) field inside the inclusion is uniform, then the inclusion must be an ellipse. This is a joint work.AN with Graeme W. Milton.