Abstract
We prove that in Minkowski spaces, a harmonic function does not necessarily satisfy the mean value formula. Conversely, we also show that a function satisfying the mean value formula is not necessarily a harmonic function. Finally, we conclude that in a Minkowski space, if all harmonic functions have the mean value property or any function satisfying the mean value formula must be a harmonic function, then the Minkowski space is Euclidean.
Highlights
Harmonic functions play a crucial role in many areas of mathematics, physics, and engineering
To give a cleaner path to the question, we list some examples above that are special cases of a Randers–Minkowski space
We can conclude that in a Minkowski space, a harmonic function has no relationship with the mean value formula, and vice versa
Summary
Harmonic functions play a crucial role in many areas of mathematics, physics, and engineering. It is well known that harmonic functions in Euclidean spaces admit the mean value property, and vice versa. Where Br ( x0 )(⊂ Ω) denotes the ball centered at x0 of radius r In this short note, we are able to discuss this issue in a Minkowski space. To compare it with the Minkowski metric, we equip it with a norm F (y) on Rn such that:. There have been many research works about harmonic functions on Riemannian manifolds. The study of harmonic functions has been developed for Finsler manifolds. For a more general harmonic map on Finsler manifolds, we refer the reader to [11]
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