Poisson’s integral formula via heat kernels
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Yoichi Miyazaki
Abstract
We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.
Acknowledgements
The author is grateful for the referee who gave valuable comments and suggestions, particularly for pointing out that the reflection method is also applicable to wedge domains, and drawing my attention to reference [1].
References
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