Solutions to the Pólya–Szegö Conjecture and the Weak Eshelby Conjecture

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, that is, that if the field inside an inclusion is uniform for all uniform loadings, then the inclusio...

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Published inArchive for rational mechanics and analysis Vol. 188; no. 1; pp. 93 - 116
Main Authors Kang, Hyeonbae, Milton, Graeme W.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 01.04.2008
Springer
Springer Nature B.V
Subjects
Classical Mechanics
Complex Systems
Exact sciences and technology
Fluid- and Aerodynamics
Fundamental areas of phenomenology (including applications)
Mathematical and Computational Physics
Mathematical methods in physics
Numerical approximation and analysis
Ordinary and partial differential equations, boundary value problems
Physics
Physics and Astronomy
Solid mechanics
Static elasticity (thermoelasticity...)
Structural and continuum mechanics
Theoretical
Polarization Tensor
Isoperimetric Inequality
Boundary Measurement
Milton
Lipschitz Boundary
Eshelby tensor
Solid inclusion
Ellipsoid
Modeling
Elliptic shape
Uniform field
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Summary: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, that is, 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 Pólya and Szegö on the isoperimetric inequalities for the polarization tensors (PTs). The Pólya–Szegö conjecture asserts that the inclusion whose electrical PT has the minimal trace takes the shape of a disk or a ball.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-007-0087-z