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


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