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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.

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