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I'm reading the first chapter of the book A geometric approach to free boundary problems by Caffarelli and Salsa, see the PDF here. The question came from Page 14—15. Let me state my question:

Let $f_{\epsilon}(s)$ be a smooth approximation of Dirac measure (more details can be found in the book), and the support of $f_{\epsilon}(s)$ is the interval $[0,\epsilon]$. Given boundary data $g \in H^1(B_1)$, $g \ge 0$, and now consider the Dirichlet type equation:

$$2 \Delta u=f_{\epsilon}(u),\,u_{\epsilon}(0)=\epsilon,\, u|_{\partial B_1}=g $$

Implicit in the chapter the authors use the existence of smooth solution of the equation above. I've never learned this before, especially there is a constraint on the origin. Can anyone give me some references? Thanks in advance!

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    $\begingroup$ 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). $\endgroup$
    – leo monsaingeon
    Jul 19, 2015 at 15:39
  • $\begingroup$ @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. $\endgroup$
    – student
    Jul 19, 2015 at 20:57

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