In this topic, we proved a very interesting property involving the Laplacian of the Kelvin transform of a function. Recall that, for a given function , its Kelvin transform is defined to be
.
We then have
where the inversion point of is defined to be
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Therefore, the Kelvin transform can be defined to be
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We also have another formula
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The right hand side of the above identity involves , we can rewrite this one in terms of . Actually, we have the following
which gives
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Today, we study the a more general form of the Kelvin transform. The above definition of the Kelvin transform is with respect to the origin and unit ball in . We are now interested in the case when the Kelvin transform is defined with respect to a fixed ball. The following result is adapted from a paper due to M.C. Leung published in Math. Ann. in 2003.
Define the reflection on the sphere with center at and radius by
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It is direct to check that
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We observe that
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The Kelvin transform with center at and radius of a function is given by
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We remark that if and are fixed, and if , then as well, where . In addition, sends a set of small diameter not too close to to a set of small diameter. We verify that double Kelvin transform is the identity map. We have
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