发表机构
Department of Mathematics, Barnard College, Columbia University; Dipartimento di Matematica Università di Bologna（哥伦比亚大学巴纳德学院数学系; 博洛尼亚大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文是献给Sandro Salsa 75岁生日的纪念卷社论，回顾了自由边界问题中黏性与变分方法的发展，包括经典问题、Alt-Caffarelli的贡献及现代扩展。
AI 中文摘要
自由边界问题（FBPs）是定义域依赖于解的微分方程，在火焰传播、金融数学和肿瘤生长等多个领域具有广泛应用。经典例子包括Bernoulli问题、障碍问题和Stefan问题。该领域通过变分和几何方法取得了显著进展，尤其是Alt、Caffarelli及其合作者，他们发展了单调性公式和黏性解来处理自由边界正则性。研究扩展已涵盖非线性算子、非齐次问题以及低维情形，如薄Bernoulli问题。演化问题，如Hele-Shaw流动，也与FBPs相关联。为庆祝这些进展并致敬Sandro Salsa的开创性贡献，2024年在巴勒莫举行的AMS-UMI联合会议上设立了FBPs专题会议。本文是献给Sandro Salsa 75岁生日的纪念卷的社论。
英文摘要
Free boundary problems (FBPs) are differential equations where the domain depends on the solution, with applications in diverse fields such as flame propagation, financial mathematics, and tumor growth. Classical examples include the Bernoulli problem, the obstacle problem, and the Stefan problem. The field has advanced significantly through variational and geometric approaches, notably by Alt, Caffarelli, and collaborators, who developed monotonicity formulas and viscosity solutions to address free boundary regularity. Extensions have included nonlinear operators, inhomogeneous problems, and lower-dimensional cases like the thin Bernoulli problem. Evolution problems, such as the Hele-Shaw flow, also connect to FBPs. To celebrate these developments and honor Sandro Salsa's pioneering contributions, a special session on FBPs was held at the AMS-UMI joint meeting in Palermo (2024). This is the editorial of a volume dedicated to Sandro Salsa on the occasion of his 75th birthday.
