Lemma 8.6.5.DFE78235C5E9 Let $u$ be bounded admissible and $h\geq u$ smooth. Choose
\[ b>\max \{ 1,\| H_ m(h)/\omega ^ n\| _\infty \} . \]
For each positive integer $N$, let $w_ N$ be the smooth strictly admissible solution of
\[ H_ m(w_ N)=b e^{N(w_ N-h)}\omega ^ n. \]
With
\[ M_ N=\| h\| _\infty +\frac{\log b}{N},\qquad E_ N=\frac{m\log (N+1)+\log b}{N}, \]
one has $\| w_ N\| _\infty \leq M_ N$ and
\[ u-E_ N-2M_ N/N\leq w_ N\leq h. \]
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
\[ T_ N=(\omega +\mathrm{dd}^{\mathrm{c}}w_ N)^{m-1}\wedge \omega ^{n-m}, \qquad d_ N=H_ m(w_ N)/\omega ^ n>0. \]
Condition (8.2.2) and polarized positivity yield, at $x_ N$,
\[ (N+1)d_ N\geq \omega \wedge T_ N/\omega ^ n \geq d_ N^{(m-1)/m}. \]
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
\[ u\leq (1+N^{-1})w_ N+E_ N+M_ N/N \leq w_ N+E_ N+2M_ N/N. \]
$\square$
Pome
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