Catherine Lebiedzik | Searcharxiv

arXiv2026

On the characterization of polyharmonic functions through iterated means

We introduce an infinite family of mean-value formulas (exact and asymptotic) given in terms of linear combinations of iterated means. We prove that the mean-value formulas in this family characterize real-valued polyharmonic functions of finite order, and that a simple algebraic condition partitions the family into equivalence classes according to the order of polyharmonicity. Our key results include strong converses to the mean-value properties -- locally integrable functions satisfying a mean-value property in the family are polyharmonic -- and a regularity result -- locally integrable functions satisfying a mean-value property in the family, whether exact or asymptotic, are smooth.

math.AP↗

arXiv2026

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

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.

math.AP↗