Saccon, Claudio On an evolution problem with free boundary. (English) Zbl 0739.49009 Houston J. Math. 16, No. 1, 87-120 (1990). From the introduction: H. W. Alt and L. A. Caffarelli [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)] have studied the regularity properties of the minima of the functional \[ f(u)=\int_ \Omega\|\nabla u(x)\|^ 2 dx+\hbox{meas}(\{x:\;u(x)>0\}) \] on the set of all \(u\) in \(H^ 1(\Omega,\mathbb{R})\) such that \(u=\varphi\) on \(\partial\Omega\) (\(\Omega\) is an open set in \(\mathbb{R}^ n\) and \(\varphi\) is fixed). In this work the evolution problem for \(f\) is studied, in the particular case of \(n=1\), \(\Omega=]0,1[\). More precisely in Theorem 3.1 it is proved that, for \(u_ 0\) in a suitable class, there exists a “curve of maximal slope for \(f\)” starting from \(u_ 0\). This curve \(U\), as a consequence of the regularity results proved in Section 2, solves a “heat equation” where positive and where negative, and verifies certain conditions (for the space derivative) on the “ free boundary” of the set \(\{U>0\}\). Moreover it is proved that the function \(f\) decreases along the curve \(U\). Reviewer: I.Athanasopoulos (Iraklion) Cited in 1 Document MSC: 49J40 Variational inequalities 35R35 Free boundary problems for PDEs Keywords:evolution problem; curve of maximal slope; heat equation; free boundary Citations:Zbl 0449.35105 PDFBibTeX XMLCite \textit{C. Saccon}, Houston J. Math. 16, No. 1, 87--120 (1990; Zbl 0739.49009)