Solutions to the conjectures of Polya-Szego and Eshelby

TL;DR

This paper proves the weak Eshelby conjecture in three dimensions, confirming that uniform internal fields imply ellipsoidal shapes, and verifies the Polya-Szego conjecture on polarization tensors, advancing understanding in elasticity and potential theory.

Contribution

It proves the weak Eshelby conjecture in three dimensions and provides an alternative proof of the strong conjecture in two dimensions, also confirming the Polya-Szego conjecture on polarization tensors.

Findings

Weak Eshelby conjecture proven in 3D.

Alternative proof of the strong Eshelby conjecture in 2D.

Polya-Szego conjecture on polarization tensors confirmed.

Abstract

Eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform. He then conjectured that the converse is true, i.e., that if the field inside an inclusion is uniform for all uniform loadings, then the inclusion is of elliptic or ellipsoidal shape. We call this the weak Eshelby conjecture. In this paper we prove this conjecture in three dimensions. In two dimensions, a stronger conjecture, which we call the strong Eshelby conjecture, has been proved: If the field inside an inclusion is uniform for a single uniform loading, then the inclusion is of elliptic shape. We give an alternative proof of Eshelby's conjecture in two dimensions using a hodographic transformation. As a consequence of the weak Eshelby's conjecture, we prove in two and three dimensions a conjecture of Polya and Szego on the isoperimetric inequalities for the polarization tensors. The Polya-Szego conjecture asserts that the inclusion whose electrical polarization tensor has the minimal trace takes the shape of a disk or a ball.