An unstable elliptic free boundary problem arising in solid combustion. (English) Zbl 1119.35123

The elliptic problem \[ \Delta u=-\chi_{\{u>0\}} \] is unstable in the sense that it corresponds to the classical obstacle problem with inverted sign. It is related to the travelling wave problem for the combustion of a solid with ignition temperature. The main results of the paper are: (i) the existence of a maximal and a minimal solution, (ii) the analyticity of a local minimizer of the energy functional out of the free boundary \(\partial \{u>0\}\) which is also locally analytic, (iii) the analysis of the possible singular points of the free boundary and of the behaviour of \(u\) near them. The two-dimensional case is studied in particular deriving some interesting information on the structure of the free boundary.


35R35 Free boundary problems for PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
80A25 Combustion
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