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The flux $F(p)=p/\sqrt{1+|p{|}^2}$ of the minimal surface equation for a graph has a positive definite symmetric matrix derivative. Its eigenvalues are $(1+|p{|}^2{)}^{-3/2}$ in the direction of $p$ and $(1+|p{|}^2{)}^{-1/2}$ orthogonally; at $p=0$ every eigenvalue is one. On $|p|\le M$ it satisfiesThus an averaged linearization of a nonlinear divergence-form equation is uniformly elliptic where the relevant gradients are bounded. Pointwise positivity alone does not supply a uniform bound on an unbounded gradient range.

For a $C^2$ solution $u$ of the minimal surface equation for a graph on the closure of a bounded domain,Write $a^{ij}(Du)u_{ij}=0$ with $a^{ij}(p)={\delta }_{ij}-p_i{p}_j/(1+|p{|}^2)$. Its differentiated operator $\mathcal{L}=a^{ij}{D}_{ij}+b^{\ell }{D}_{\ell }$, where $b^{\ell }=(\partial a^{ij}/\partial p_{\ell })u_{ij}$, satisfiesThe weak maximum principle for elliptic operators gives the result. Interior Schauder estimates justify differentiating a solution initially assumed only $C^2$.

If a smooth minimal surface equation for a graph solution on ${\mathbb{R}}^n\setminus \{0\}$ is homogeneous of degree one, with $n\ge 2$, then its graph extends to a plane through zero. Its gradient is homogeneous of degree zero, so $|Du{|}^2$ attains a global maximum on the unit sphere. The strong maximum principle for elliptic operators makes it constant. The differentiated equation in the gradient maximum principle for a minimal graph then forces $D^{2}u=0$, and degree-one homogeneity removes the affine constant. This argument applies to cones that are graphs of smooth functions away from their vertex; it is not a claim that every minimal surface cone is flat.

Write the minimal surface equation for a graph as $\mathrm{div}F(Du)=0$ in its weak formulation, where $F(p)=p/\sqrt{1+|p{|}^2}$. For negative increments use the same difference quotient convention, soFor $\zeta \in C_c^1({\Omega }^{\prime \prime })$, the function ${\delta }_{\ell ,h}\zeta$ is an admissible test function in $\Omega$. Commuting the first partial derivative with translation, then applying discrete integration by parts, givesThe fundamental theorem of calculus along a line segment gives the averaged linearization of a nonlinear divergence-form equationThus the backward difference quotient solves a linear equation in divergence form:To verify the ellipticity of the minimal surface flux, computeand henceThe eigenvalue parallel to $p$ is $(1+|p{|}^2{)}^{-3/2}$; every orthogonal eigenvalue is $(1+|p{|}^2{)}^{-1/2}$. The same lower and upper bounds pass to the averaged symmetric matrix $A_h$ whenever both endpoint gradients have norm at most $M$:The printed $C^1(\Omega )$ hypothesis supplies such an $M$ on each relatively compact subset, uniformly for sufficiently small $h$. It therefore establishes a locally uniformly elliptic operator. A single lower bound on all of ${\Omega }^{\prime \prime }$ additionally requires bounded gradients there and at the shifted points; this is automatic for bounded $\Omega$ and fixed $h$, but is not supplied on an arbitrary open set.

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:

Let $u=v+\psi$, so $v$ has zero Dirichlet boundary condition. Using the supplied form of the minimal surface equation for a graph, setFor $\Vert \psi {\Vert }_{C^{2,\alpha }}\le \beta \le 1$ and $\Vert v\Vert ,\Vert w\Vert \le 1$, the difference bound impliesandThe shifted arguments stay in the permitted radius-two ball, and $\Vert f_{\psi }{\Vert }_{C^{0,\alpha }}\le C{\beta }^2+C_n\beta$. The Laplacian with zero Dirichlet boundary condition has trivial null space by the maximum principle for harmonic functions. Choose $\beta >0$ small enough that $2C\beta \le \delta$ and $C{\beta }^2+C_n\beta \le {\epsilon }_0$, for the constants in part (c) with $L=\Delta$. Its small-data existence for a nonlinear elliptic Dirichlet problem supplies $v$, and $\boxed{u=v+\psi }$ solves the minimal surface equation for a graph with the required boundary values.

For a compactly supported variation $v+t\varphi$, differentiation of the area functional at $t=0$ gives the first variationAfter integration by parts, this is the minimal surface equation for a graphFor $v_R(x)=R^{-1}v(Rx)$ one has $D{v}_R(x)=Dv(Rx)$. The equation is invariant under this scaling, so $v_R$ solves it on $B_1$ for every $R>0$.

The first variation of the graph area functional isThus the minimal surface equation for a graph isor, in nondivergence form,Differentiate the divergence equation with respect to $x_k$. Then $w=D_{k}u$ satisfieswhere

The inverse first-fundamental-form matrix isThus $2H=\mathrm{tr}((I^S{)}^{-1}I{I}^S)$ vanishes exactly whenThis is the minimal surface equation for a graph.

Part c supplies a bounded slope condition constant $K$. Choose $k>K$. Parts a and d give a constrained minimizer with $\mathrm{Lip}(u)\le K<k$, and part b makes it a minimizer over all of ${\mathrm{Lip}}_g(B_1)$.

For $F(\xi )=\sqrt{1+|\xi {|}^2}$, the Euler-Lagrange equation is the minimal surface equation for a graphThe Lipschitz bound confines $Du$ to a compact set on which $D^{2}F$ is uniformly positive definite, so the equation is uniformly elliptic. Interior regularity gives $C^{1,\alpha }$ first, and repeated Schauder estimates then give smoothness in the interior.