 Schauder estimates

Schauder estimates control Hölder norms of derivatives of a solution to an elliptic differential equation. For a second-order uniformly elliptic operator $L=a^{ij}{D}_{ij}+b^i{D}_i+c$ with $C^{0,\alpha }$ coefficients, where $0<\alpha <1$, interior estimates gain two derivatives over a $C^{0,\alpha }$ forcing, while estimates up to a boundary also require regular boundary data.

On a bounded $C^{2,\alpha }$ domain, a standard global Dirichlet problem estimate isHere $g$ denotes a $C^{2,\alpha }$ extension of the prescribed boundary data. For homogeneous data choose $g=0$. The constant depends on ellipticity, domain geometry, coefficient bounds and $\alpha$; it is independent of the particular solution. The $C^0$ term allows a nontrivial homogeneous kernel and can be removed only with an additional uniqueness estimate.^[ref]

The boundary term is essential. On the unit disk, $u_n(r,\theta )=r^n\cos (n\theta )$ is harmonic with supremum one but boundary second tangential derivative of size $n^2$. Its global $C^{2,\alpha }$ norm cannot be bounded using only zero forcing and that fixed supremum. An interior Schauder estimate remains valid on a smaller disk, whose distance from the boundary controls the constant. This separates interior regularity from control of rapidly oscillating boundary data.