×

On an evolution problem with free boundary. (English) Zbl 0739.49009

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 \(U\).

MSC:

49J40 Variational inequalities
35R35 Free boundary problems for PDEs

Citations:

Zbl 0449.35105
PDFBibTeX XMLCite