PomeCorollary 12.10.21.D179EF5E5E25 Under the hypotheses of proposition 12.10.20, suppose in addition that the obstacle envelope $P_{\delta \omega }(u_0)$ of section 5.40 is not identically $-\infty $ for some real $0<\delta <1$. Then $v_\lambda \to u_0$ in capacity as $\lambda \downarrow 0$. In particular this holds if the threshold solution $u_0$ is bounded.
Corollary 12.10.21D179EF5E5E25
Proof. Fix a sequence $\lambda _ j\downarrow 0$. Proposition 12.10.20 gives $u_0-C\leq v_{\lambda _ j}\leq 0$ and $v_{\lambda _ j}\to u_0$ in $L^1$. For all $j$ with $\lambda _ j\leq 1$,
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (v_{\lambda _ j}) =c_{\lambda _ j}\mathrm{e}^{\lambda _ jv_{\lambda _ j}}f\, \mathrm{d}V \leq b\mathrm{e}^{C}f\, \mathrm{d}V. \]
The bounded functions $|\mathrm{e}^{v_{\lambda _ j}}-\mathrm{e}^{u_0}|$ tend to zero in $\mathrm{d}V$-measure. Since $f\in L^1$, their integrals against $f\, \mathrm{d}V$ tend to zero, so the error term in (15.9.2) tends to zero. We apply proposition 15.9.3 with $\theta =\omega $ and $\varphi =u_0$; its additional hypothesis that the $L^1$ limit $u$ satisfies $u\preceq \varphi $, in the notation of definition 16.7.1, holds trivially, since $u=u_0$. This proves convergence in capacity along every sequence, and hence for the family. Finally, if $u_0$ is bounded, a constant below it is a competitor for $P_{\delta \omega }(u_0)$, so this envelope is not identically $-\infty $. $\square$
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