Abstract
In this paper, for the Green’s function of a bounded convex domain \(\Omega \) with pole at \(x_0\in \Omega \), we find the auxiliary curvature function which satisfies a differential inequality and so attains its minimum on the boundary. Moreover, we obtain a convexity estimate for the Green’s function of the domain above and give the proof of its specific convexity from the viewpoint of partial differential equations themselves.
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Acknowledgments
The author would like to express sincere gratitude to Prof. Xi-Nan Ma for his encouragement and many discussions in this subject. The author would also like to thank the referee for his (her) very careful reading and many good suggestions on this paper. Research of the author was supported in part by NSFC(No.11326144) and the Foundation of Harbin Normal University (No.KGB201224).
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Communicated by F. H. Lin.
