A minimization problem and the regularity of solutions in the presence of a free boundary.

*(English)*Zbl 0545.35013The 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)].

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