PomeProposition 8.7.4.26DD802A141E For every smooth real obstacle $h$, the envelope $P(h)$ equals the pointwise supremum of all bounded admissible functions below $h$.
Proposition 8.7.426DD802A141E
Proof. The solutions $w_ N$ of 8.6.5 are continuous admissible competitors, so $w_ N\leq P(h)$. For each bounded admissible $u\leq h$, the same lemma gives
\[ u\leq w_ N+E_ N+2M_ N/N\leq P(h)+E_ N+2M_ N/N. \]
Let $N\to \infty $ for this fixed obstacle and fixed $b$. Thus $P(h)$ dominates every bounded competitor. Since $P(h)$ itself is a bounded competitor, it equals their supremum. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.