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The Boundary Schauder estimate iswhere $C$ depends only on the dimension, $\alpha$, the ellipticity constants, and the coefficient Hölder norms.

Subtract a $C^{2,\alpha }$ extension of $\phi$ to reduce to a function $v$ with zero data on the flat boundary; this changes the forcing by a controlled $C^{0,\alpha }$ term. The key local estimate is that for every $\delta >0$,To prove it, argue by contradiction. A failing normalized sequence has points $x_k,y_k$ at which the second-derivative Hölder quotient stays nonzero. Set $r_k=|x_k-y_k|$, subtract the appropriate second-order Taylor polynomial, and rescale space by $r_k$ and the functions by $r_k^{2+\alpha }[D^2{v}_k{]}_{\alpha }$. Necessarily $r_k\to 0$.

If the rescaled distance to the flat boundary tends to infinity, the domains converge to all of ${\mathbb{R}}^n$; otherwise they converge to a half-space. Coefficient compactness freezes the principal matrix, while the normalized right-hand sides converge locally uniformly to zero. A linear change of variables turns every limiting equation into the Laplace equation. In the whole-space case the Polynomial-growth Liouville theorem for harmonic functions makes the limit a polynomial of degree at most two. In the half-space case the zero boundary data permit odd reflection across the flat boundary, after which the same theorem applies. The subtracted Taylor normalization forces that polynomial to vanish to second order, contradicting the nonzero limiting Hölder oscillation.

Apply the local estimate on all interior balls and boundary half-balls. The interior and boundary forms of the Simon absorption lemma absorb the term multiplied by $\delta$, while the Hölder interpolation inequality absorbs the remaining lower $C^2$ norm into the top seminorm and the $C^0$ norm. Restoring $\phi$ yields the displayed estimate.

Solved by gpt-5.6-sol high.

Iteration gives the interior derivative estimate for a harmonic functionThe growth hypothesis bounds the right side by $C^{\prime }{R}^{\alpha -1}+o(1)$, which tends to zero as $R\to \infty$. Thus every derivative of order $k+1$ vanishes everywhere. The Taylor theorem makes $u$ a polynomial of degree at most $k$, proving the Polynomial-growth Liouville theorem for harmonic functions.

Solved by gpt-5.6-sol high.