内容来源：https://www.quantamagazine.org/long-sought-proof-tames-some-of-maths-unruliest-equations-20260206/

内容总结：

百年数学难题终获突破：意大利学者攻克非均匀椭圆偏微分方程正则性理论

偏微分方程（PDE）是描述自然界动态过程的核心数学工具，从气象预测到金融模型，再到疾病传播，其应用无处不在。然而，绝大多数描述真实复杂系统的偏微分方程因其极端复杂性，长期以来无法被严格分析和求解。数学家们通常退而求其次：不求精确解，但求证明解是“正则”的——即解的行为平滑、稳定，没有突兀的跳跃，从而确保近似求解的有效性。但一类至关重要的方程始终是理论盲区：描述非均匀材料（如成分复杂的熔岩、复合建筑材料、生物组织）的“非均匀椭圆偏微分方程”。自上世纪30年代波兰数学家尤利乌什·肖德尔建立经典正则性理论以来，该理论能否适用于这类更贴近现实世界的方程，成了悬置百年的关键难题。

近日，这一百年壁垒终于被打破。意大利帕尔马大学的朱塞佩·明乔内教授与其博士生克里斯蒂安娜·德·菲利皮斯合作，成功将肖德尔理论拓展至非均匀椭圆方程，完成了该领域一项里程碑式的工作。他们的论文已于去年夏季正式发表，为科学家精确描述众多长期无法数学建模的真实自然现象打开了大门。

突破之路：从“时间机器”到“绝望中的奇迹”

明乔内早在2000年，时年28岁，在俄罗斯一次学术会议上便意识到经典理论对非均匀情形可能失效。回国后，他与同事提出猜想：要保证非均匀方程的解正则，除了经典条件，还必须附加一个与材料非均匀程度严格相关的“不等式”条件来控制方程系数的变化。然而，证明这一猜想异常艰难，明乔内多年尝试未果后一度放弃。

转机出现在2017年。当时的一年级博士生德·菲利皮斯不顾旁人劝阻，主动联系明乔内，决心挑战这个沉寂近二十年的难题。“这就像一台时间机器，”明乔内回忆道，“仿佛遇到了20年前的自己来叩响我的思维之门。”德·菲利皮斯的新鲜视角与执着热情促使两人重启研究。

关键创新：“幽灵方程”与毫厘之争

证明的核心在于控制解的变化速度（即梯度）不致无限增大。他们从原方程构造出一个“幽灵方程”，以此作为间接分析的桥梁。德·菲利皮斯提出关键思路，通过一套冗长而精妙的步骤，从“幽灵方程”中成功“打捞”出梯度信息。

随后，他们将梯度拆解为无数细微部分，并证明每一部分的大小均受到严格约束。这个过程容不得丝毫误差，“是一场永无止境的游戏”，德·菲利皮斯形容。经过无数次推倒重来，他们最终在2022年的预印本中完成了绝大部分证明，并在后续工作中补全了最后一块拼图，精确验证了明乔内当年提出的不等式正是正则性与非正则性之间的分水岭。明乔内将这一成果称为“绝望中的奇迹”。

深远影响：为理解真实世界打开数学之窗

此项突破不仅完成了一项延续百年的数学计划，其更重要的意义在于为科学研究提供了强有力的新工具。以往，为了数学处理方便，科学家们往往被迫将非均匀材料假设为均匀的，从而使用过度简化的模型。现在，数学家们可以直接研究描述真实复杂介质的方程。

此外，该证明所发展的方法有望被推广至其他类型的偏微分方程，包括同时随时间与空间变化的方程。赫尔辛基大学的图奥莫·库西评价道：“神奇之处在于，他们将众多深刻理论整合到一个框架下，并最终挤压出了证明。”正如德·菲利皮斯所言，在这一数学突破的背后，“是一个等待被解释的广阔现实世界”。

中文翻译：

千呼万唤始出来：数学证明驯服部分最棘手的方程

引言

风暴的轨迹、股价的演变、疾病的传播——数学家可以使用所谓的偏微分方程来描述任何随时间或空间变化的现象。但问题在于：这些“PDE”通常过于复杂，以至于无法直接求解。

数学家转而依赖一种巧妙的变通方法。他们可能不知道如何计算给定方程的精确解，但他们可以尝试证明这个解必须是“正则的”，或者说在某种意义上是行为良好的——例如，其数值不会以物理上不可能的方式突然跳跃。如果一个解是正则的，数学家就可以使用各种工具来近似它，从而更好地理解他们想要研究的现象。

然而，许多描述实际情况的PDE仍然遥不可及。数学家一直未能证明它们的解是正则的。特别是，这些难以企及的方程中，有一部分属于一类特殊的PDE，研究人员花了一个世纪来发展其理论——但这一理论却始终无法适用于这个子类。他们碰壁了。

如今，两位意大利数学家终于取得了突破，将该理论扩展到了那些更混乱的PDE。他们去年夏天发表的论文，标志着一个雄心勃勃项目的顶峰，首次使得科学家能够描述那些长期以来抗拒数学分析的真实现象。

“捣蛋鬼”还是“乖孩子”？

火山喷发时，炽热、混乱的熔岩流在地面奔涌。但数小时或数天（甚至更久）之后，它会冷却到足以进入一种平衡状态。虽然熔岩覆盖的广阔区域内温度仍因地而异，但其温度已不再时刻变化。

数学家使用所谓的椭圆型PDE来描述此类情况。这些方程表示在空间上变化但不随时间变化的现象，例如流经岩石的水压、桥梁上的应力分布，或肿瘤中营养物质的扩散。

但椭圆型PDE的解很复杂。例如，熔岩PDE的解在给定一些初始条件下，描述了其在每一点的温度。它依赖于许多相互作用的变量。

研究人员希望即使无法写出这样的解，也能近似它。但他们使用的方法只有在解是正则的情况下才能很好地工作——这意味着解没有任何突然的跳跃或扭结。（熔岩的温度不会在各地出现尖锐的峰值。）“如果出了问题，很可能是因为缺乏正则性，”里斯本大学的马克森·桑托斯说。

20世纪30年代，波兰数学家尤利乌什·绍德尔试图确立椭圆型PDE必须满足的、能保证其解正则的最低条件。他证明，在许多情况下，你只需证明嵌入方程中的规则——例如热量在熔岩中传播速度的规则——不会在点与点之间变化得太突然。

自绍德尔的证明以来，几十年来，数学家们已经证明，这个条件足以确保任何描述一种“均匀”材料的PDE都有正则解。在这种材料中，基本规则的变化幅度是有限的。例如，如果你假设熔岩是均匀的，热量总是在一定的速度限制内流动，永远不会太快或太慢。

但熔岩实际上是熔融岩石、溶解气体和晶体的混合物。在这种非均匀材料中，你无法控制极端情况，热量传播速度可能因位置不同而产生更剧烈的差异：熔岩中的某些区域导热性可能极好，而另一些区域则极差。在这种情况下，你将使用“非均匀椭圆型”PDE来描述这种情况。

几十年来，没有人能够证明绍德尔的理论适用于这类PDE。

不幸的是，“现实世界是非均匀椭圆的，”意大利帕尔马大学的数学家朱塞佩·明焦内说。这意味着数学家们陷入了困境。明焦内想弄清楚原因。

时光机

2000年8月，28岁、刚刚获得博士学位的明焦内来到俄罗斯一个破旧的旧度假村，参加一个关于微分方程的会议。一天晚上，无事可做，他开始阅读在旅途中结识的数学家瓦西里·瓦西里耶维奇·日科夫的论文，他意识到，那些看似行为良好的非均匀椭圆型PDE，即使在满足绍德尔所确定的条件时，也可能有不正则的解。绍德尔的理论在非均匀情况下不仅仅是更难证明，它需要更新。

回到意大利后，他与两位同事合作，提出非均匀椭圆型PDE需要满足一个额外的条件才能保证其解正则。不仅控制热流的规则必须逐点逐渐变化，而且这些变化必须受到严格控制，以应对熔岩的非均匀性。特别是，数学家们假设，材料越不均匀，这种控制就必须越严格。他们将这个条件表示为一个不等式，给出了系统可以容忍的非均匀程度的精确阈值。

他们证明，对于不等式不成立的PDE，他们无法再保证解是正则的。但他们无法证明这个不等式精确地标定了解从正则变为可能不正则的临界点。明焦内花了数年时间研究这个问题，但徒劳无功。他最终放弃了努力。

近二十年过去了。然后在2017年，一位名叫克里斯蒂安娜·德·菲利皮斯的一年级研究生听说了将绍德尔理论扩展到非均匀椭圆型方程的努力。更有经验的数学家警告她不要研究这个问题，但她无视了他们的建议，联系了明焦内。在一次深夜的Skype通话中，她告诉他，她对于如何证明他的猜想有一些想法，并决心从他中断的地方继续下去。

“这就像一台时光机，”明焦内说。“就像遇到了二十年前的自己，敲响了我自己思想的门。”

据他说，是德·菲利皮斯的“新的能量、热情和相信这件事可以完成的信念”说服了他，让他重新拾起了这个长期休眠的、证明自己猜想的尝试。

奇迹

证明PDE的解是正则的关键在于表明它总是以受控的方式变化。数学家通过观察一个描述解在每一点变化速度的特殊函数来做到这一点。他们想证明这个被称为梯度的函数不会变得太大。

但正如通常无法直接计算PDE的解一样，通常也无法计算其梯度。

相反，德·菲利皮斯和明焦内从原始PDE推导出了他们所谓的“幽灵方程”，这是他们真正所需之物的一个影子。

这正是明焦内几十年前卡住的地方。但德·菲利皮斯有一个想法，可以改进幽灵方程，使其能更清晰地反映PDE。通过一个漫长、多步骤的过程，两人能够从幽灵方程中获得足够的信息来恢复梯度。

“这样做有点牵强，”德国比勒费尔德大学的西蒙·诺瓦克说。“但它奏效了，而且相当漂亮。”

现在他们必须弄清楚如何证明他们恢复的梯度不会变得太大。他们将其分割成更小的部分，并证明每个部分都不能超过特定的大小。这付出了巨大的努力：即使单个部分出现微小的测量误差，也会影响他们对梯度的估计，使他们偏离想要证明的阈值。

在2022年的一篇预印本中，他们能够很好地控制所有这些部分，从而证明大多数满足明焦内不等式的非均匀椭圆型PDE必须具有正则解。但一些PDE仍然缺失。为了证明完整的猜想，数学家们必须对梯度各部分的大小给出更好的界。绝对没有回旋余地。这需要多次重新开始——“一场永无止境的游戏，”德·菲利皮斯说。但最终，他们能够证明明焦内几十年前预测的阈值是完全正确的。

他说，这是“绝望中的奇迹”。

德·菲利皮斯和明焦内不仅完成了一个长达一个世纪的项目。他们还使得数学家能够研究复杂的现实过程，而这些过程直到现在都不得不使用不切实际的简化方程来建模。

研究人员也兴奋地希望应用他们的技术来理解其他类型的偏微分方程，包括那些在空间和时间上都变化的方程。“神奇之处在于，他们将所有这些深刻的理论汇集到一个框架下，然后挤压出了证明，”赫尔辛基大学的图奥莫·库西说。

PDE在数学分析上一直几乎困难得令人望而却步。现在它们变得稍微容易了一点。在它们背后，德·菲利皮斯说，“有一个巨大的现实”等待着被解释。

英文来源：

Long-Sought Proof Tames Some of Math’s Unruliest Equations
Introduction
The trajectory of a storm, the evolution of stock prices, the spread of disease — mathematicians can describe any phenomenon that changes in time or space using what are known as partial differential equations. But there’s a problem: These “PDEs” are often so complicated that it’s impossible to solve them directly.
Mathematicians instead rely on a clever workaround. They might not know how to compute the exact solution to a given equation, but they can try to show that this solution must be “regular,” or well-behaved in a certain sense — that its values won’t suddenly jump in a physically impossible way, for instance. If a solution is regular, mathematicians can use a variety of tools to approximate it, gaining a better understanding of the phenomenon they want to study.
But many of the PDEs that describe realistic situations have remained out of reach. Mathematicians haven’t been able to show that their solutions are regular. In particular, some of these out-of-reach equations belong to a special class of PDEs that researchers spent a century developing a theory of — a theory that no one could get to work for this one subclass. They’d hit a wall.
Now, two Italian mathematicians have finally broken through, extending the theory to cover those messier PDEs. Their paper, published last summer, marks the culmination of an ambitious project that, for the first time, will allow scientists to describe real-life phenomena that have long defied mathematical analysis.
Naughty or Nice
During a volcanic eruption, a scorching, chaotic river of lava flows over the ground. But after hours or days (or perhaps even longer), it cools enough to enter a state of equilibrium. Its temperature is no longer changing from moment to moment, although it still varies from place to place across the vast expanse of space the lava covers.
Mathematicians describe situations like this using what are called elliptic PDEs. These equations represent phenomena that vary across space but not time, such as the pressure of water flowing through rock, the distribution of stress on a bridge, or the diffusion of nutrients in a tumor.
But solutions to elliptic PDEs are complicated. The solution to the lava PDE, for instance, describes its temperature at every point, given some initial conditions. It depends on a lot of interacting variables.
Researchers want to approximate such a solution even when it’s impossible to write it down. But the methods they use only work well if the solution is regular — meaning that it doesn’t have any sudden jumps or kinks. (There won’t be sharp spikes in the lava’s temperature from place to place.) “If something goes wrong, it’s probably because of the [lack of] regularity,” said Makson Santos of the University of Lisbon.
In the 1930s, the Polish mathematician Juliusz Schauder sought to establish the minimal conditions an elliptic PDE must satisfy to guarantee that its solutions will be regular. He showed that in many cases, all you have to prove is that the rules baked into the equation — such as the rule for how quickly heat will spread in lava — do not change too abruptly from point to point.
In the decades since Schauder’s proof, mathematicians have shown that this condition is enough to ensure that any PDE that describes a nice, “uniform” material has regular solutions. In such a material, there’s a limit on how extreme the underlying rules can get. For example, if you assume your lava is uniform, heat will always flow within certain speed limits, never too quickly or slowly.
But lava is actually a diverse mix of molten rock, dissolved gases, and crystals. In such a nonuniform material, you can’t control the extremes, and you might get more drastic differences in how quickly heat can spread, depending on your location: Some regions in the lava might conduct heat extremely well, and others extremely poorly. In this case, you’ll use a “nonuniformly elliptic” PDE to describe the situation.
For decades, no one could prove that Schauder’s theory held for this kind of PDE.
Unfortunately, “the real world is nonuniformly elliptic,” said Giuseppe Mingione, a mathematician at the University of Parma in Italy. That meant mathematicians were stuck. Mingione wanted to understand why.
Time Machine
In August 2000, Mingione — 28 years old and fresh off his Ph.D. — found himself in a dilapidated old resort in Russia, attending a conference on differential equations. One evening, with nothing better to do, he started reading papers by Vasiliĭ Vasil’evich Zhikov, a mathematician he’d met on the trip, and he realized that nonuniformly elliptic PDEs that seem well behaved can have irregular solutions even when they satisfy the condition Schauder had identified. Schauder’s theory wasn’t simply harder to prove in the nonuniform case. It needed an update.
Giampiero Palatucci
Returning to Italy, he teamed up with two colleagues and proposed that nonuniformly elliptic PDEs needed to satisfy an additional condition to guarantee that their solutions would be regular. Not only did the rules governing heat flow have to change gradually from point to point, but these changes had to be tightly controlled to account for the lava’s nonuniformity. In particular, the mathematicians posited, the more uneven the material, the tighter this control would have to be. They represented this condition as an inequality, giving a precise threshold for how much nonuniformity a system could tolerate.
They showed that for PDEs where the inequality did not hold, they could no longer guarantee that the solutions would be regular. But they couldn’t prove that the inequality precisely marked the point where solutions would go from being regular to potentially irregular. Mingione spent years on the problem, to no avail. He eventually abandoned the effort.
Almost 20 years passed. Then in 2017, a first-year graduate student named Cristiana De Filippis heard about the quest to extend Schauder’s theory to nonuniformly elliptic equations. More experienced mathematicians warned her against pursuing the problem, but she ignored their advice and reached out to Mingione. Over a late-night Skype call, she told him that she had some ideas for how to prove his conjecture and was determined to pick up where he had left off.
Giampiero Palatucci
“It was like a time machine,” Mingione said. “It was like meeting myself of 20 years before and knocking at the door of my own mind.”
According to him, it was De Filippis’ “new energy and enthusiasm and faith that this could be done” that persuaded him to revive his long-dormant attempt to prove his conjecture.
Miracles
The key to proving that the solution to a PDE is regular is to show that it always changes in a controlled way. Mathematicians do this by looking at a special function that describes how fast the solution changes at each point. They want to show that this function, which is called the gradient, can’t get too big.
But just as it’s usually impossible to directly compute the solution to a PDE, it’s also usually impossible to calculate its gradient.
Instead, De Filippis and Mingione derived what they called a “ghost equation” from the original PDE, a shadow of what they actually needed.
This was where Mingione had gotten stuck decades earlier. But De Filippis had an idea for how to hone the ghost equation so that it could give a crisper view of the PDE. Using a long, multistep procedure, the pair was able to gain enough information from the ghost equation to recover the gradient.
“It’s kind of far-fetched to do it like this,” said Simon Nowak of Bielefeld University in Germany. “But it works, and it’s quite beautiful.”
Now they had to figure out how to show that the gradient they’d recovered couldn’t get too large. They split it into smaller pieces and proved that each piece couldn’t exceed a specific size. This took an enormous amount of effort: Even a tiny measurement error on a single piece would throw off their estimate of the gradient, taking them away from the threshold they were aiming to prove.
In a 2022 preprint, they were able to tame all these pieces well enough to show that most nonuniformly elliptic PDEs that satisfy Mingione’s inequality have to have regular solutions. But some PDEs were still missing. To prove the full conjecture, the mathematicians had to get even better bounds on the sizes of the gradient’s pieces. There was absolutely no wiggle room. This required starting over many times — “a never-ending game,” De Filippis said. But eventually, they were able to prove that the threshold Mingione had predicted decades earlier was exactly right.
It was “a miracle by desperation,” he said.
De Filippis and Mingione haven’t just completed a century-long project. They’ve also made it possible for mathematicians to study complicated real-life processes that until now had to be modeled using unrealistically simplified equations.
Researchers are also excited to apply their techniques to understand other kinds of partial differential equations, including ones that change in both space and time. “The magical part is that they were bringing all this deep theory under one umbrella and then squeezing out the proof,” said Tuomo Kuusi of the University of Helsinki.
PDEs have always been almost prohibitively difficult to analyze mathematically. Now they’ve gotten just a little bit easier. Behind them, De Filippis said, “there is an enormous reality” waiting to be explained.