Solutions to the Pólya–Szegö Conjecture and the Weak Eshelby Conjecture
Solutions to the Pólya–Szegö Conjecture and the Weak Eshelby Conjecture
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DOI：
10.1007/s00205-007-0087-z
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发表时间：
2008-04
影响因子：
2.5
通讯作者：
Hyeonbae Kang;G. Milton
中科院分区：
文献类型：
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作者：
Hyeonbae Kang;G. Milton
EShelby证明，如果夹杂物是椭圆形或椭球状的，那么对于任何均匀的弹性载荷，夹杂物内部的场是均匀的。然后他猜想反之亦然，即如果一个夹杂内的场对于所有均匀载荷都是均匀的，那么这个夹杂是椭圆形或椭球状的。我们称其为弱EShelby猜想。在本文中，我们从三维角度证明了这一猜想。在二维情况下，证明了一个更强的猜想，我们称之为强EShelby猜想：如果对于单一均匀载荷，夹杂内部的场是均匀的，则夹杂是椭圆形的。我们用热图变换给出了二维EShelby猜想的另一种证明。作为弱EShelby猜想的结果，我们在二维和三维上证明了Pólya和Szegö关于极化张量(PTS)的等周不等式的猜想。Pólya-Szegö猜想断言，其电子PT具有最小踪迹的包裹体的形状为圆盘或球状。
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.