PomeLemma 8.7.3.C39ADEAC74E3 The function $P(h)$ is continuous and admissible, and $H_ m(P(h))$ is supported on $\{ P(h)=h\} $. Moreover $\| P(h_1)-P(h_2)\| _\infty \leq \| h_1-h_2\| _\infty $.
Lemma 8.7.3C39ADEAC74E3
Proof. For smooth $h$, use the penalty solutions $w_ N$ in 8.6.2. Its maximum argument applies to every continuous admissible competitor $v\leq h$, with an error depending on $h$ and $N$, and not on $v$. Taking their supremum gives
\[ P(h)-E_ N-2M_ N/N\leq w_ N\leq P(h), \]
where the error tends to zero. Thus $P(h)$ is their uniform limit and is continuous and admissible. On every compact subset of $\{ P(h)<h\} $ the penalty densities $b e^{N(w_ N-h)}$ tend uniformly to zero. Product continuity implies the contact support assertion. Adding constants to competitors gives the stated Lipschitz bound. Approximate a continuous $h$ uniformly by smooth obstacles. Their envelopes converge uniformly to $P(h)$, proving continuity and admissibility. Any compact subset of $\{ P(h)<h\} $ also lies outside the contact sets of all sufficiently close smooth obstacles; product continuity gives the support assertion for $h$. $\square$
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