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A compactness proof of the interior Schauder estimate can magnify a pair of points where the Hölder seminorm of the Hessian matrix is nearly attained. If the proposed estimate fails, normalize that seminorm to one while lower norms and forcing tend to zero. The separation $r_j$ of the chosen pair tends to zero. Subtract the quadratic Taylor polynomial and divide by $r_j^{2+\alpha }$ after rescaling space by $r_j$.

The resulting functions have uniformly bounded Hessian Hölder seminorms on expanding domains and vanishing Taylor data at zero. Compact embedding of Hölder spaces yields an entire harmonic function whose second derivatives have finite global $\alpha$-Hölder seminorms, with $0<\alpha <1$. The polynomial-growth Liouville theorem for harmonic functions makes those derivatives constant, contradicting the normalized nonzero oscillation. The Simon absorption lemma removes the residual small multiple of the larger-ball seminorm.

Integration by parts gives the minimal surface equation for a graph in nondivergence form:Its eigenvalues are between $(1+|p{|}^2{)}^{-1}$ and one. On each relatively compact subdomain, $Du$ is bounded, so it is a uniformly elliptic operator. Since $u\in C^2$, its coefficient functions are locally Lipschitz. The interior Schauder estimate gives $C_{\mathrm{loc}}^{2,\beta }$ regularity for every $0<\beta <1$; differentiating and repeatedly applying regularity estimates gives $\boxed{u\in C^{\infty }(\Omega )}$.

Differentiate in direction $k$. Each $u_k$ satisfies $\mathcal{L}{u}_k=0$, withFor $v=|Du{|}^2$ this yieldsIf $u\in C^2(\overline{\Omega })$, the principal coefficients are uniformly elliptic globally and the drift is bounded. The weak maximum principle for elliptic operators bounds $v$ by its boundary maximum. Boundary values are limits of interior values by continuity, proving the gradient maximum principle for a minimal graph:

The interior Schauder estimate iswhere $C$ depends only on $n$, $\alpha$, the ellipticity constant, and the stated coefficient bounds.

It is enough first to prove the estimate for the Hölder seminorm of $D^{2}u$. The Interpolation inequality in Holder spaces controls $Du$ and $D^{2}u$ by a small multiple of $[D^{2}u{]}_{\alpha }$ plus $\Vert u{\Vert }_0$; the small term is later absorbed. Freezing $a^{ij}$ at the centre of each ball and moving coefficient differences and lower-order terms to the right reduces the local estimate toOnce this is known, the Simon absorption lemma gives the desired estimate on $B_{1/2}$.

For completeness, prove the frozen-coefficient estimate by contradiction. If it failed, choose solutions $u_k$, points $x_k$, and scales ${\rho }_k\downarrow 0$ at which the scale-invariant Hölder quotient is almost maximal. Let $q_k$ be the quadratic Taylor polynomial of $u_k$ at $x_k$ and defineThen $v_k(0)=D{v}_k(0)=D^2{v}_k(0)=0$, the Hessians have uniformly bounded local $C^{0,\alpha }$ seminorms, and the normalization makes their oscillation nonzero on a fixed ball. The localized equation iswhere the coefficient matrices converge locally uniformly to one constant positive-definite matrix and the given normalized error $g_k$ tends locally uniformly to zero.

The Arzelà-Ascoli theorem produces a locally $C^2$ convergent subsequence with limit $v$ satisfying a constant-coefficient elliptic equation on ${\mathbb{R}}^n$. A linear rotation and scaling turn it into a harmonic function. The normalization gives $D^{2}v(0)=0$ but a nonconstant Hessian, while maximality of the scaled quotient gives growth at most $C(1+|y{|}^{2+\alpha })$. Applying the Liouville theorem to derivatives shows that every second derivative is constant because $0<\alpha <1$. This contradicts the normalized Hessian oscillation. The frozen estimate follows, and interpolation plus absorption completes the proof.

Put $h=\log X$. The first equation becomeswhile the second remainsThe assumed $C^{0,\alpha }$ regularity and the bounds away from zero make $X^{-2}$ a $C^{0,\alpha }$ uniformly elliptic coefficient. The divergence-form $C^{1,\alpha }$ estimate from part (d) first gives $Y\in C_{\mathrm{loc}}^{1,\alpha }$. Hence the right side of the equation for $h$ is $C^{0,\alpha }$, and the interior Schauder estimate gives $h$, and therefore $X$, in $C_{\mathrm{loc}}^{2,\alpha }$.

Expanding the $Y$ equation givesThe higher-order Schauder estimates now alternate between the equations for $h$ and $Y$, gaining derivatives at each step. Induction yields $(X,Y)\in C^{\infty }({\Omega }^{\prime };H)$ for every ${\Omega }^{\prime }⋐\Omega$.

The function $u$ solves the linear nondivergence equationThe coefficient matrix is uniformly elliptic, and part (ii) makes it $C^{0,\alpha }$ on every smaller ball. Solve the corresponding linear Dirichlet problem on $B_r⋐B_1$ with boundary value $u$. Existence gives a $C^{2,\alpha }$ solution, and the weak maximum principle for elliptic operators gives uniqueness, so that solution is $u$. Hence $u\in C_{\mathrm{loc}}^{2,\alpha }(B_1)$.

The local boundedness estimate for uniformly elliptic equations first controls $\Vert u{\Vert }_{C^0(B_r)}$ by $\Vert u{\Vert }_{L^2(B_1)}$. Applying the interior Schauder estimate and covering $B_{\theta }$ by finitely many balls gives

For every ${\Omega }^{\prime }⋐\Omega$, the interior Schauder estimate isThe constant depends on $n$, $\alpha$, the ellipticity constants, the coefficient $C^{0,\alpha }$ norms, $\Omega$, ${\Omega }^{\prime }$, and in particular the distance from ${\Omega }^{\prime }$ to the boundary. It is independent of $u$ and $f$.