Theorem 14.19.51979B56319AA1

Proof. The right-hand sides in the family with $\| \partial h\| _\infty \leq H_1$ form an equicontinuous set. By corollary 14.19.46, the corresponding functions $U$ form a precompact subset of $C^0(X)$. Choose a finite $\eta $-net in this set, consisting of smooth admissible functions $V_1,\ldots ,V_ N$. Each solution is within $\eta $ of some $V_ j$, so proposition 14.19.50 bounds its gradient by $C(1+\max _ j\| \partial V_ j\| _\infty ^2)$. The set of solutions, and hence the chosen finite net, depends only on the fixed data and $H_1$. Finally, $u=2\pi U$, so the same conclusion holds for $\mathrm du$. $\square$