Polya-Szego和Eshelby的解
Solutions to the conjectures of Polya-Szego and Eshelby
摘要：
Eshelby表明，如果夹杂物是椭圆形或椭球形，那么对于任何均匀的弹性载荷，夹杂物内部的场是均匀的。他接着又说，反之亦然，即，如果夹杂内部的场对所有均匀载荷都是均匀的，则夹杂是椭圆形或椭球形的。我们称之为弱Eshelby猜想。在本文中，我们证明了这个猜想在三维空间。在二维空间中，证明了一个更强的猜想，我们称之为强Eshelby猜想：如果包含体内部的场在单一均匀载荷下是均匀的，则该包含体是椭圆形的。我们给出了另一种证明Eshelby猜想在二维使用速端变换。作为弱Eshelby猜想的结果，我们在二维和三维空间中证明了Polya和Szego关于极化张量的等周不等式的一个猜想. Polya-Szego猜想认为，其电极化张量具有最小迹线的包裹体具有圆盘或球的形状。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
