Notes on Schauder estimates by scaling for elliptic PDEs in divergence form
Stefano Vita
TL;DR
These notes develop a self-contained local Schauder theory for weak solutions of $-\mathrm{div}(A\nabla u)=f+\mathrm{div}F$ in $B_1$ with uniformly elliptic $A$. Employing a scaling-regularization-blow-up scheme inspired by Simon $(Sim97)$ and leveraging compactness and Liouville-type rigidity, they establish a full ladder of regularity results, from a priori $C^{0,\alpha}$ and $C^{1,\alpha}$ estimates to $H^2$ and $H^k$ regularity for smooth data, and then derive a posteriori $C^{0,\alpha}$, $C^{1,\alpha}$, and $C^{k,\alpha}$ estimates via a mollification-approximation framework. The treatment emphasizes geometric-analytic techniques and connects interior Schauder theory to free boundary problems through blow-up arguments. A key feature is that the presence of the field term $F$ enables bootstrap to higher regularity by iterating $C^{1,\alpha}$ estimates, while the modular approach accommodates both continuity and Hölder continuity of coefficients. The results provide sharp, scale-aware interior regularity for divergence-form elliptic equations and offer a bridge to geometric PDE applications and obstacle-type problems.
Abstract
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.