AI 中文总结
研究高阶泊松方程解的均值公式，通过迭代均值线性组合给出，余项由强迫项振荡量化，证明正则性结果与均值性质强逆命题，为高阶泊松方程解提供均值刻画。
AI 中文摘要
高阶泊松方程涉及拉普拉斯算子的整数幂和非零强迫项，出现在流体动力学、结构工程和图像处理等物理与工程领域。我们引入了一族高阶泊松方程解的均值公式，由迭代均值的线性组合给出，其精确余项由强迫项的振荡量化。我们还证明了一个正则性结果以及均值性质的强逆命题，即仅满足我们公式的局部可积函数是高阶泊松方程的正则解。与齐次情形不同，这里的正则性由强迫项的正则性决定。我们的结果共同给出了高阶泊松方程解的均值刻画。
英文摘要
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.
