Abstract
In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain on which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap intersecting \(\mathbb {S}^{n-1}\) orthogonally. As an application, we show that a stationary point for a partially torsional rigidity under a volume constraint must be a spherical cap.
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Acknowledgements
We are indebted to Professor Guofang Wang for numerous insight and discussion on this topic. We also thank Professor Martin Man-Chun Li for his interest. We are grateful to the anonymous referee who attracts our attention to the existence problem and the excellent book [13] on the shape optimization, as well as his/her numerous suggestions which help to improve the paper considerably.
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Communicated by J. Jost.
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This work is supported by NSFC (Grant No. 11501480, 11871406) and the Natural Science Foundation of Fujian Province of China (Grant No. 2017J06003) and the Fundamental Research Funds for the Central Universities (Grant No. 20720180009).
