Theorem 16.16.74B6432E0991B

Proof. First let $r=1$, and suppose that $f=f_1$ is smooth. By lemma 16.16.1, the solutions $a_\beta $ have uniformly bounded values, gradients and full real Hessians for $\beta \geq \beta _0$, and by proposition 16.15.21 they converge uniformly to $P_\kappa (f)$. The limit $P_\kappa (f)$ is therefore continuous, and by the same proposition it has bounded metric Laplacian; hence it belongs to $C^{1,\alpha }(X)$ by proposition 16.14.9, and in particular it is $C^1$.

We now pass to the limit in the Hessian bounds. For a smooth vector field $Y$, the function $\nabla ^2a_\beta (Y,Y)=YYa_\beta -(\nabla _ YY)a_\beta $ is obtained from $a_\beta $ by a second-order differential operator. Since $a_\beta \to P_\kappa (f)$ uniformly, $\nabla ^2a_\beta (Y,Y)\to \nabla ^2P_\kappa (f)(Y,Y)$ in the sense of distributions, where the right side is the weak covariant Hessian, and for every nonnegative test function $\varphi $ we obtain $|\langle \nabla ^2P_\kappa (f)(Y,Y),\varphi \rangle |\leq C\int |Y|_{g_0}^2\varphi $. By corollary 2.8.2(1), the weak quadratic components $\nabla ^2P_\kappa (f)(Y,Y)$ are bounded functions, with $|\nabla ^2P_\kappa (f)(Y,Y)|\leq C|Y|_{g_0}^2$ almost everywhere; by polarization, so are all components of the weak covariant Hessian.

The gradient is treated in the same way: $Ya_\beta \to YP_\kappa (f)$ in the sense of distributions, and $|Ya_\beta |\leq C|Y|_{g_0}$; since $YP_\kappa (f)$ is continuous, pairing with nonnegative test functions of integral one supported in small balls about a point shows that $|YP_\kappa (f)|\leq C|Y|_{g_0}$ everywhere, so $|\nabla P_\kappa (f)|_{g_0}\leq C$. In a chart, the weak second derivatives of $P_\kappa (f)$ differ from the components of its weak covariant Hessian by the Christoffel symbols times its first derivatives, so they are bounded on compact subsets of the chart. Hence $P_\kappa (f)$ is $C^{1,1}$, by proposition 2.8.5, and the bounds just proved show that its $C^{1,1}$ norm is bounded by a constant depending only on the fixed geometry and on the constant $B$ of lemma 16.16.1 for $f$.

Now let the $f_ j$ be arbitrary $C^{1,1}$ functions. Using convolution in finitely many charts and a smooth partition of unity, we obtain smooth functions $f_{j,\rho }$ that converge to $f_ j$ in $C^1$ as $\rho \to 0$ and have uniformly bounded covariant Hessians. Indeed, the second derivatives of such a partitioned convolution are the cutoffs times the convolutions of the bounded weak second derivatives of $f_ j$, plus terms that involve only the values and first derivatives of the local convolutions; all of these are uniformly bounded. For $\delta >0$ put

The weights $p_ j$ are positive with sum one, $\min _ jf_{j,\rho }-\delta \log r\leq F_{\rho ,\delta }\leq \min _ jf_{j,\rho }$, and the gradient of $F_{\rho ,\delta }$ is the convex combination $\sum _ jp_ j\nabla f_{j,\rho }$. For every real tangent vector $Y$ the chain rule gives

because the term in parentheses is a variance and hence nonnegative. Let $\rho ,\delta \to 0$ with $\delta \log r\leq 1$. Then the smooth obstacles $F_{\rho ,\delta }$ have uniformly bounded values and gradients and a uniform upper bound for their Hessians, although their Hessians have no uniform lower bound as $\delta \to 0$; this is why lemma 16.16.1 uses only an upper bound. Since the smooth case bounds the $C^{1,1}$ norm of the envelope in terms of the fixed geometry and the constant $B$ of that lemma, the envelopes $P_\kappa (F_{\rho ,\delta })$ have uniformly bounded $C^{1,1}$ norms. The obstacles converge uniformly to $\min _ jf_ j$, so the envelopes converge uniformly to $P_\kappa (\min _ jf_ j)$ by the stability estimate of proposition 16.11.4(4). The limit is $C^1$ by theorem 16.15.24, since $C^{1,1}$ functions have bounded metric Laplacian. The same passage to the limit as in the smooth case, with the smooth functions $a_\beta $ replaced by the $C^{1,1}$ functions $P_\kappa (F_{\rho ,\delta })$, whose gradients and weak covariant Hessians are bounded by a uniform constant, now proves the theorem, together with the bound for the norm. $\square$