2006/09/13 by Hyeonbae Kang, Graeme W. Milton, Kang, Hyeonbae +1
Materials Science · Mathematics · #30E25 #31B10 #31B20 #35R35 #74B05 #74B10 #74N15 #74P10 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Finite Group Theory Research #Optimization and Control (math.OC) #Quasicrystal Structures and Properties #math.AP #math.OC #msc:30E25 #msc:31B10 #msc:31B20 #msc:35R35 #msc:74B05 #msc:74B10 #msc:74N15 #msc:74P10
paper · pdf · doi:10.48550/arxiv.math/0609374
22 pages, 1 figure
arxiv created 2006/09/13 · openalex publication_date 2006/09/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.