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Suppose $v_k\to v$ in $C^{1,\beta }$ and write $u_k=T(v_k)$. The preceding uniform Schauder estimate bounds $(u_k)$ in $C^{2,\alpha \beta }$. Every subsequence therefore has a further subsequence converging in $C^{1,\beta }$ by the compact embedding of Hölder spaces. The composed coefficients converge uniformly, and lower-exponent Hölder compactness provides enough convergence of the second derivatives to pass to the linear equation. Every subsequential limit solves the problem defining $T(v)$.

Uniqueness of that linear Dirichlet problem forces every such limit to equal $T(v)$. Since every subsequence has a further subsequence with this same limit, the whole sequence converges to $T(v)$ in $C^{1,\beta }$. Thus $T$ is continuous.

Solved by gpt-5.6-sol high.

Join the Laplace operator to the target operator byThe family has one ellipticity constant. The global Schauder estimate and the maximum principle give, uniformly in $t$,for zero boundary data. Let $I$ contain those $t$ for which $L_t:C_0^{2,\alpha }\to C^{\alpha }$ is onto. The assumed Laplace solvability gives $0\in I$; the bounded inverse theorem and small perturbations make $I$ open; and the uniform estimate plus compactness of lower Hölder embeddings makes $I$ closed. The method of continuity yields $I=[0,1]$. At $t=1$ this gives the required solution, and the maximum principle gives uniqueness.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.