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Jul 19, 2015 at 20:57 comment added student @leo monsaingeon, thanks very much! I thought the assumption in Theorem 1.2 was imposed everywhere later in the chapter otherwise the proofs wouldn't work, but if taking them as renormalized results, then I understand them much better.
Jul 19, 2015 at 15:39 comment added leo monsaingeon You can show smoothness by local interior regularity: since $f_{\epsilon}$ is bounded you have $u\in W^{2,p}_{loc}$ for any large $p$, thus by Morrey's inequality $u\in C^{1,\alpha}_{loc}$. Bootstrapping on Schauder's estimates you immediately get $u\in \mathcal{C}^{\infty}$ (of course not all the way up to the boundary). As far as I can tell the $u(0)=\epsilon$ is not imposed, so it's not a constraint (cf the "suppose" in their theorem 1.2) but only a normalization (which they explain just below the theorem).
Jul 19, 2015 at 0:48 history edited student CC BY-SA 3.0
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Jul 18, 2015 at 21:48 history edited Joonas Ilmavirta CC BY-SA 3.0
Reformatted the first paragraph.
Jul 18, 2015 at 21:08 history edited student CC BY-SA 3.0
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Jul 18, 2015 at 20:54 history asked student CC BY-SA 3.0