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Discrete Morse semiflows of a functional with free boundary. (English) Zbl 0795.35150
For each \(n=1,2,3,\dots\), let \[ J_ n(u)=\int_ \Omega\left( {{| u- u_{n-1}|^ 2} \over h} +|\nabla u|^ 2+ Q^ 2(x) \chi(u>0)\right)d{\mathcal L}^ m,\tag{1} \] where \(h>0\), \(\Omega\) is a domain in \(\mathbb{R}^ n\) with Lipschitz boundary, \(u:\Omega\to \mathbb{R}\) a suitable real valued function, \(\chi\) denotes characteristic function and \({\mathcal L}^ m\) is the \(m\)-dimensional Lebesgue measure.
The authors show how to define \(u_ n\) recursively (as being minimizers of (1) under certain kind of free boundary condition) and observe that the same sequence satisfies the time discretization of the heat equation. Hence they call \(u_ n\) the discrete Morse semiflow. Certain subsequences converge to a function \(u_ \infty\) which is a local minimizer of (1) with \(n=\infty\). They also show that the boundary of \(u_ \infty>0\) has locally finite perimeter. They mention previous results by Alt, Caffarelli and Friedman about existence of global minimizers (as a free boundary problem) of the functional \[ J(u)= \int_ \Omega (F(|\nabla u|^ 2)+ Q^ 2(x) \chi(u>0)) d{\mathcal L}^ m\tag{2} \] and claim that the procedure in the present paper leads to finding also minimizers of (2) which are not global.

35R35 Free boundary problems for PDEs
35J20 Variational methods for second-order elliptic equations
35A15 Variational methods applied to PDEs
49J40 Variational inequalities