A minimization problem and the regularity of solutions in the presence of a free boundary. (English) Zbl 0545.35013

Phillips, Daniel

Indiana Univ. Math. J. 32, 1-17 (1983).

The author deals with the minimization problem \[ J(u)=\int_{G}(| \nabla u|^ 2/2+u^{\gamma})dx=\min_{w\in K}J(w),\quad u\in K, \] where G is an open bounded set in \(R^ n\) with Lipschitz boundary \(\partial G\), \(0<\gamma<1\) a fixed constant and \(K=\{w\in H^ 1(G):w\geq 0,\) \(w=u_ 0|_{\partial G}\}\) with \(u_ 0\in H^ 1(G)\) given, \(u_ 0\geq 0\). It is shown that at least one minimizer \(u\in K\) exists. Also u is shown to be subharmonic in G. The main result of the paper is the interior regularity of u, i.e. for any compact subset \(H\subset G\), \(u\in C^{1,\beta -1}(H)\) with \(\beta =2/(2-\gamma).\)
An interesting ingredient in the proof is the observation that the functional J preserves minimizers under a particular scaling. In the extreme cases \(\gamma =1\) and \(\gamma =0\) the same problem was previously studied by L. A. Caffarelli [Commun. Partial Differ. Equations 5, 427-448 (1980; Zbl 0437.35070)] and by H. Alt and L. A. Caffarelli [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)].

Reviewer: V.Mustonen

MSC:

35B65	Smoothness and regularity of solutions to PDEs
35A15	Variational methods applied to PDEs
35R35	Free boundary problems for PDEs
49J20	Existence theories for optimal control problems involving partial differential equations

Keywords:

subharmonic minimizer; minimization problem; interior regularity