Three spheres inequality

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In mathematics, the three spheres inequality bounds the norm of an harmonic function on a given sphere in terms of the norm of this function on two spheres, one with bigger radius and one with smaller radius.

Mathematics field of study concerning quantity, patterns and change

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Harmonic function function with vanishing Laplacian

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Statement of the three spheres inequality

Let be an harmonic function on . Then for all one has

where for is the sphere of radius centred at the origin and where

Here we use the following normalisation for the norm:

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References

Digital object identifier Character string used as a permanent identifier for a digital object, in a format controlled by the International DOI Foundation

In computing, a Digital Object Identifier or DOI is a persistent identifier or handle used to uniquely identify objects, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.

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