标题： 关于通过缩放的Schauder估计的笔记，用于散度形式的椭圆偏微分方程 显示英文标题

标题： Notes on Schauder estimates by scaling for elliptic PDEs in divergence form

摘要： 这些是2025年春季在帕维亚举行的博士课程“自由边界问题和椭圆偏微分方程的正则性”的部分讲义。 目标是通过西蒙的工作精神，以缩放的方式提供对散度形式的二阶线性椭圆偏微分方程的经典内部和局部Schauder估计的全面且自洽的处理。 这里介绍的主要技术具有几何性质，并主要是在研究如极小曲面等几何问题时发展起来的。 采用的方法依赖于紧性及爆破论证，结合刚性结果（Liouville定理），并与其他部分博士课程中研究自由边界问题所使用的方法有许多相似之处，那部分课程的主要主题是自由边界问题。 显示英文摘要

摘要： These are the notes of a part of the PhD course Regularity for free boundary problems and for elliptic PDEs, held in Pavia in the spring of 2025. The aim is to provide a comprehensive and self-contained treatment of classical interior and local Schauder estimates for second-order linear elliptic PDEs in divergence form via scaling in the spirit of Simon's work. The main techniques presented here are geometric in nature and were primarily developed in the study of geometric problems such as minimal surfaces. The adopted approach relies on compactness and blow-up arguments, combined with rigidity results (Liouville theorems), and shares many features with the one used in the study of free boundary problems, which was the main topic of the other part of the PhD course.