两位意大利数学家在 2025 年夏发表的论文，解决了一个困扰数学界近百年的难题：证明一类“非均匀椭圆型”偏微分方程（PDE）的解具有正则性。偏微分方程可描述随时间或空间变化的现象，但往往无法直接求解。若能证明解不会出现不现实的跳变或奇点，数学家就能近似分析真实系统。此前，这一方法在一大类现实相关方程上失效，形成长期理论空白。

椭圆型 PDE 描述空间变化而非时间演化的系统，例如桥梁应力、岩石渗流或熔岩冷却。1930 年代，波兰数学家 Schauder 证明，只要方程中的“规则”在空间中变化不剧烈，解就会是正则的。随后数十年，这一结论被证明适用于“均匀”材料，即物性参数有上下界。然而现实材料往往高度非均匀，导热或扩散能力在不同位置可相差极大，对应的非均匀椭圆型 PDE 无法满足既有理论假设。

突破源于对反例的系统理解。2000 年 8 月，年仅 28 岁的 Giuseppe Mingione 发现，即便满足 Schauder 的条件，某些非均匀椭圆型 PDE 仍可能产生不规则解，说明理论本身需要修正而非简单推广。经过二十多年发展，新结果首次将正则性理论扩展到这类方程，使数学分析能够覆盖更多真实世界系统，并为物理、生物和工程中的复杂模型提供可靠基础。

A paper published in the summer of 2025 by two Italian mathematicians resolved a problem that had challenged mathematics for nearly a century: proving regularity for a class of nonuniformly elliptic partial differential equations (PDEs). PDEs describe phenomena that change in time or space but are often impossible to solve exactly. If a solution can be shown to avoid unphysical jumps or singularities, it becomes accessible to approximation. Until now, this approach failed for a broad set of realistic equations, leaving a major theoretical gap.

Elliptic PDEs model spatial variation without time evolution, such as stress on a bridge, fluid flow through rock, or cooling lava. In the 1930s, the Polish mathematician Schauder proved that if the rules encoded in an equation vary smoothly in space, its solutions are regular. Over subsequent decades, this result was extended to “uniform” materials whose properties stay within fixed bounds. Real materials, however, are often highly nonuniform, with transport rates that can differ drastically by location, placing them outside the reach of the classical theory.

The breakthrough came from understanding counterexamples. In August 2000, Giuseppe Mingione, then 28 years old, realized that some nonuniformly elliptic PDEs can have irregular solutions even when they satisfy Schauder’s conditions, implying the theory itself needed revision. After more than two decades of work, the new proof finally extends regularity theory to these equations, enabling mathematical analysis of many real-world systems and strengthening models across physics, biology, and engineering.