Corollary 12.10.312A3A58A80CB4

Proof. By proposition 15.28.7, the constant $c$ for which $\mathrm{MA}_\omega (v)=cf\, \mathrm{d}V$ has a continuous solution is $b_\omega (f)$. The continuous normalized solution is therefore a threshold solution, and it equals $u_0$ by the uniqueness in theorem 12.10.17. For $L^ p$ data the greatest exponential solution coincides with the bounded solution, as explained after theorem 15.35.2. By proposition 15.20.1, applied with the trivial group, the normalized family is uniformly precompact, and every cluster limit solves the threshold equation. Uniqueness identifies every cluster limit with $u_0$, which proves uniform convergence of the whole family. $\square$