A two phase elliptic singular perturbation problem with a forcing term
The authors study the following two phase elliptic singular perturbation problem \[ \Delta u^\varepsilon =\beta_\varepsilon(u^\varepsilon) +f^\varepsilon\quad \text{in}\;\Omega\subset \mathbb R^N, \] where \(\varepsilon>0,\) \(\beta_\varepsilon(s)={1\over \varepsilon} \beta\left({s\over \varepsilon}\right)\) with a Lipschitz continuous function \(\beta,\) satisfying \(\beta>0\) in \((0,1),\) \(\beta\equiv0\) outside \((0,1)\) and such that \(\int \beta(s)\,ds=M,\) and the functions \(u^\varepsilon\) and \(f^\varepsilon\) are supposed to be uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit in the presence of sources. Uniform estimates and passing to the limit when \(\varepsilon\to0\) enable the authors to show that the limit functions are solutions of the two phase free boundary problem \[ \begin{aligned} &\Delta u= f\chi_{\{u\not\equiv0\}}\quad \text{in}\;\Omega\setminus \partial\{u>0\},\cr &| \nabla u^+| ^2-| \nabla u^-| ^2=2M\quad \text{on}\;\Omega\cap \partial\{u>0\}, \end{aligned} \] where \(f=\lim f^\varepsilon\) in a viscosity sense, and in pointwise sense at the regular free boundary points. It is shown in addition that the free boundary is smooth whence, under appropriate assumptions, the limit functions are classical solutions to the considered free boundary problem. Some of the results are new even in the case \(f^\varepsilon\equiv0.\) The results at the paper apply also to other combustion models, such as ones with nonlocal diffusion and/or transport. Several of these applications are discussed and, in some cases full regularity of the free boundary is obtained.
scientific article; zbMATH DE number 5125018
scientific article