Radial Symmetry for Elliptic Boundary-Value Problems on Exterior Domains

• Published in: Archive for rational mechanics and analysis Vol. 137; no. 4; pp. 381 - 394
• Main Author: Reichel, Wolfgang
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 26.06.1997
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s002050050034

By the Alexandroff-Serrin method [2, 14] of moving hyperplanes we obtain radial symmetry for the domain and the solutions of (ProQuest: Formulae and/or non-USASCII text omitted; see image) on an exterior domain (ProQuest: Formulae and/or non-USASCII text omitted; see image) , subject to the overdetermined boundary conditions (ProQuest: Formulae and/or non-USASCII text omitted; see image) , (ProQuest: Formulae and/or non-USASCII text omitted; see image) on (ProQuest: Formulae and/or non-USASCII text omitted; see image) , (ProQuest: Formulae and/or non-USASCII text omitted; see image) at (ProQuest: Formulae and/or non-USASCII text omitted; see image) and (ProQuest: Formulae and/or non-USASCII text omitted; see image) in (ProQuest: Formulae and/or non-USASCII text omitted; see image) . In particular, the following conjecture from potential theory due to P. Gruber (cf. [11, 8]) is proved: Let (ProQuest: Formulae and/or non-USASCII text omitted; see image) or (ProQuest: Formulae and/or non-USASCII text omitted; see image) be a bounded smooth domain with a constant source distribution on (ProQuest: Formulae and/or non-USASCII text omitted; see image) and let (ProQuest: Formulae and/or non-USASCII text omitted; see image) be the induced single-layer potential. If (ProQuest: Formulae and/or non-USASCII text omitted; see image) is constant in (ProQuest: Formulae and/or non-USASCII text omitted; see image) , then (ProQuest: Formulae and/or non-USASCII text omitted; see image) is a ball.[PUBLICATION ABSTRACT]