8.6 Proper equations and admissible approximationA493740DE457

Proof. At a maximum of $w$, the admissible form $\omega +\mathrm{dd}^{\mathrm{c}}w$ is at most $\omega $, hence $e^{aw+H}\leq 1$ there. At a minimum it is at least $\omega $, and the reverse inequality holds. These give the bounds. For two solutions, a positive maximum of their difference would imply $H_ m(w_1)\leq H_ m(w_2)$ there, contradicting the exponential factor. This proves uniqueness.

First take $a=1$ and use the path $H_ m(w_ t)=e^{w_ t+tH}\omega ^ n$, starting with $w_0=0$. The trace estimate of 8.4.6 applies with prescribed potential $v=0$, fixed smooth comparison $0$, and $h(x,v)=tH(x)$; its constants are uniform for $t\in [0,1]$. 8.3.3 then gives a uniform gradient and complex-Hessian bound. 8.5.1 gives uniform $W^{2,p}$ and hence $C^{1,\alpha }$ bounds, for any fixed $\alpha <1$. Consequently $r_ t=e^{(w_ t+tH)/m}$ has a uniform Hölder bound.

The extension of 8.5.3 is uniformly elliptic and concave on all real Hessians, with coefficient oscillation bounded by $C|x-y|(1+\| R\| )$. The analytic estimate recalled in section 8.5 therefore gives uniform $C^{2,\beta }$ bounds on a finite cover. The original smooth Hessian operator is uniformly elliptic near the attained compact cone set. Differentiating and applying Schauder estimates gives bounds for every higher derivative.

The linearization of $\log (H_ m(w)/\omega ^ n)-w-tH$ is $L_ w-1$, where

Its principal coefficients are positive definite. The maximum principle gives zero kernel; smooth elliptic Fredholm theory gives surjectivity and Schauder inversion. The implicit-function theorem proves openness. The uniform estimates prove closedness, so the path reaches $t=1$. For general $a$, use the metric $a\omega $ and potential $\widehat w=aw$. Both sides of its coefficient-one equation acquire the same factor $a^ n$, so it is exactly the stated equation after division by that factor. $\square$

Proof. Fix a smooth $h\geq \varphi $ and a constant

For each positive integer $N$ solve, by the preceding lemma,

At a positive maximum of $w_ N-h$, the form $\omega +\mathrm{dd}^{\mathrm{c}}h$ lies above the strictly admissible form $\omega +\mathrm{dd}^{\mathrm{c}}w_ N$, hence is admissible there. Monotonicity would give $b e^{N(w_ N-h)}\leq H_ m(h)/\omega ^ n<b$, a contradiction. Thus $w_ N\leq h$. The preceding zero-order estimate also gives

Let $x_ N$ maximize $\varphi -(1+N^{-1})w_ N$. The smooth positive form $T_ N=(\omega +\mathrm{dd}^{\mathrm{c}}w_ N)^{m-1}\wedge \omega ^{n-m}$ defines a strictly elliptic linear operator. Admissibility gives $(\omega +\mathrm{dd}^{\mathrm{c}}\varphi )\wedge T_ N\geq 0$ in distributions. Its linear maximum principle, applied to the upper test $(1+N^{-1})w_ N$ plus the maximum value, gives at $x_ N$

This application is to a continuous distributional subsolution of a smooth strictly elliptic operator. It does not assume a second derivative of $\varphi $. Writing $d_ N=H_ m(w_ N)/\omega ^ n$, polarized Gårding positivity yields

Therefore, at this maximum point,

Since $\varphi \leq h$, the maximum of $\varphi -(1+N^{-1})w_ N$ is at most $E_ N+M_ N/N$. It follows on all of $X$ that

Given $\delta >0$, choose an ordinary smooth majorant $\varphi \leq h\leq \varphi +\delta $, and then $N$ so large that $E_ N+2M_ N/N\leq \delta $. The resulting $w_ N$ is strictly admissible and within $\delta $ of $\varphi $. Let $\delta \downarrow 0$. No monotonicity of the chosen approximants is needed. $\square$

Proof. Fix a Riemannian distance $d$ on $X$ and put

Each $c_ j$ is Lipschitz, $c_ j\geq u$, and $c_{j+1}\leq c_ j$. The supremum is attained. If $y_ j$ attains it at a fixed $x$, then $j d(x,y_ j)\leq \sup _ X u-u(x)$, so $y_ j\to x$. Upper semicontinuity gives $\limsup _ j c_ j(x)\leq u(x)$, and hence $c_ j\downarrow u$ pointwise.

Set $\delta _ j=4^{-j}$. Smooth approximation of continuous functions on a compact manifold gives $s_ j\in C^\infty (X)$ with $\| s_ j-c_ j\| _\infty \leq \delta _ j/4$. Take $h_ j=s_ j+\delta _ j$. Then

The bounds also imply $h_ j(x)\to u(x)$ at every point. $\square$

Proof. Existence follows from 8.6.1. The maximum comparison with $h$ in the proof of 8.6.2 gives $w_ N\leq h$; it involves only the smooth obstacle and solution. The zero-order bound from 8.6.1 gives $\| w_ N\| _\infty \leq M_ N$.

The upper semicontinuous function $u-(1+N^{-1})w_ N$ attains its maximum at some $x_ N$. The smooth function $(1+N^{-1})w_ N$ plus this maximum is an upper test for $u$ there. Put

Condition (8.2.2) and polarized positivity yield, at $x_ N$,

Consequently $d_ N\geq (N+1)^{-m}$ and $w_ N-h\geq -E_ N$ there. Using $u\leq h$, the maximum is at most $E_ N+M_ N/N$. On all of $X$ this gives

Proof. Choose smooth $h_ k\downarrow u$ by 8.6.4 and put $\varepsilon _ j=8^{-j}$. We construct smooth strictly admissible $u_ j$ with $u_ j\geq u+\varepsilon _ j$. For $j=1$, take $k_1=1$. At a subsequent step choose $k_ j\geq \max \{ j,k_{j-1}+1\} $ so large that

Such an index exists: the open sets $\{ h_ k<u_{j-1}-\varepsilon _{j-1}/2\} $ increase with $k$ and cover $X$, since $h_ k\downarrow u\leq u_{j-1}-\varepsilon _{j-1}$. Compactness gives a single index for all of $X$.

For this fixed obstacle $h_{k_ j}$ choose $b_ j$ as in 8.6.5, and then choose $N_ j$ so large that its error $E_{N_ j}+2M_{N_ j}/N_ j$ is at most $\varepsilon _ j$. The resulting smooth solution $w_ j$ satisfies

Put $u_ j=w_ j+2\varepsilon _ j$. Adding a constant preserves strict admissibility, and $u_ j\geq u+\varepsilon _ j$. For $j\geq 2$,

Finally, $u\leq u_ j\leq h_{k_ j}+2\varepsilon _ j$ and $k_ j\geq j$ give pointwise convergence to $u$. The bounds $\inf _ X u\leq u_ j\leq \sup _ X u_1$ give a common uniform bound. $\square$