## Uniform estimates for regularization of free boundary problems.(English)Zbl 0702.35252

Analysis and partial differential equations, Coll. Pap. dedic. Mischa Cotlar, Lect. Notes Pure Appl. Math. 122, 567-619 (1990).
The authors consider problems of the type $Lu=a_{ij}(x)u_{ij}+b_ i(x)u_ i+c(x)u=\beta_{\epsilon}(u)\text{ in } \Omega$ where the nonlinearity $$\beta_{\epsilon}$$ has support in [0,$$\epsilon$$ ] and $$\beta_{\epsilon}\leq B/\epsilon$$. An example of such a nonlinearity is $$\beta_{\epsilon}(u)=(1/\epsilon)b(u/\epsilon)$$, where $$\beta$$ is continuous with support in [0,1], positive on (0,1) and $$\int^{1}_{0}\beta (s)ds=M>0$$. They obtain estimates up to the boundary. For the behavior near the boundary it is assumed that u satisfies $$\mu (x)\cdot \nabla u=0$$ on $$\partial \Omega$$, $$\mu$$ pointing outside. The paper contains the following sections: a Harnack inequality up to the boundary; uniform Lipschitz continuity of $$u_{\epsilon}$$ on compact subsets of $$\Omega$$, independent of $$\epsilon$$ ; non degeneracy of certain minimal solutions; study of $$\lim_{\epsilon \to 0}u_{\epsilon}$$; study of the regularity of the free boundary; application of the results to the free boundary in a flame propagation problem. Among many other interesting results they show that $$v=\lim_{\epsilon \to 0}u_{\epsilon}$$ satisfies $$a_{ij}\nu_ i\nu_ j| \nabla v|^ 2=2M$$ on $$\sigma$$. Here is a smooth portion of the free boundary and $$v>0$$ on one side of $$\sigma$$ and $$v=0$$ on the other side.
Reviewer: R.Sperb

### MSC:

 35R35 Free boundary problems for PDEs 35B45 A priori estimates in context of PDEs