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arXiv · 2607.20776

Pointwise mean-value formulas with quantitative remainder for higher-order Poisson equations

Fernando Charro·Sophia Hertrich·Catherine Lebiedzik
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Abstract

Higher-order Poisson equations involve an integer power of the Laplacian and a nonzero forcing term, and appear in areas of physics and engineering such as hydrodynamics, structural engineering, and image processing. We introduce a family of mean-value formulas for solutions to higher-order Poisson equations, given in terms of linear combinations of iterated means, with an exact remainder quantified by the oscillation of the forcing term. We also prove a regularity result and a strong converse to the mean-value property, in the sense that merely locally integrable functions satisfying our formulas are regular solutions of the higher-order Poisson equation. In contrast with the homogeneous case, where the mean-value property forces smoothness, the regularity attainable here is dictated by the regularity of the forcing term. Together, our results provide a mean-value characterization of solutions to higher-order Poisson equations.

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Fernando Charro, Sophia Hertrich, Catherine Lebiedzik. 2026-07-22. Pointwise mean-value formulas with quantitative remainder for higher-order Poisson equations. https://arxiv.org/abs/2607.20776

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