= Schauder estimates
{c}
{wiki}
= Schauder estimate
{c}
{synonym}
<Schauder estimates> control <Hölder norms> of derivatives of a solution to an elliptic <differential equation>. For a second-order <uniformly elliptic operator> $L=a^{ij}D_{ij}+b^iD_i+c$ 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 is
$$
\|u\|_{C^{2,\alpha}(\overline\Omega)}\le C\left(\|Lu\|_{C^{0,\alpha}(\overline\Omega)}+\|u\|_{C^0(\overline\Omega)}+\|g\|_{C^{2,\alpha}(\overline\Omega)}\right),\qquad u|_{\partial\Omega}=g|_{\partial\Omega}.
$$
Here $g$ 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.<Lecture notes on Elliptic Partial Differential Equations by André Guerra and Gerard Orriols>[§§4.4–4.5, Corollary 4.19 and the homogeneous estimate (4.26)]{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.
Back to article page
