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On optimal regularity of free boundary problems and a conjecture of De Giorgi. (English) Zbl 1082.35168

The paper addresses the question of the regularity of the boundary in a very general class of free boundary problems for elliptic systems. More specifically, the authors consider \[ \begin{aligned} F_k(x,\mathbf{u},D\mathbf{u},D^2\mathbf{u})=0&\quad \text{in }\;\Omega,\;1\leq k\leq n,\tag{1}\\ G_h(x,\mathbf{u},D\mathbf{u},\nu)=0&\quad \text{on }\;\Gamma,\;1\leq h\leq\mu,\tag{2}\\ H(x,\mathbf{u},D\mathbf{u},\nu,D_{\tau}\nu)=0&\quad\text{on }\;\Gamma,\tag{3} \end{aligned} \] with \(\Omega\subset\mathbb R^N\) with the assumptions that the system (1) is elliptic, the conditions (2) are complementing for (1) (both definitions are recalled in the paper). In (2), (3) \(\Gamma\subset\partial\Omega\) denotes the free boundary, \(\nu\) is the normal to \(\Gamma, D_{\tau}\nu\) is the tangential gradient. The function \(H\) is such that (3) is a transmission condition (involving a special kind of dependence on \(D_{\tau}\nu\)). For instance the scalar one-phase problem \[ \begin{aligned} \Delta u=g(x,u,\nabla u)&\quad\text{in }\;\Omega,\\ \frac{\partial u}{\partial v}=b(x,u)&\quad\text{on }\;\Gamma,\tag{4}\\ K=h(x,u,\nabla u)&\quad\text{on }\;\Gamma, \end{aligned} \] where \(K\) is the mean curvature of \(\Gamma\), and its two-phase analog fall in this category.
The main result of the paper consists in the determination of the optimal regularity of \((u,\Gamma)\), depending on the regularity of \(F_k, G_h, H\) (up to analyticity). On the basis of their result the authors are able to answer a conjecture of De Giorgi about the analyticity of the Mumford-Shah functional. The paper provides a significant step forward in this difficult subject. The result is applicable to a great variety of situations. The method extends to cases not included in the basic theorem discussed above. In order to show this additional advantage, in the last section of the paper the authors consider the stationary Navier-Stokes equation with a free capillary surface \(\Gamma\), showing that it is enough to suppose that the velocity is \(C^1(\overline{\Omega})\), the pressure in \(C^{1,\alpha}(\overline{\Omega})\), and that \(\Gamma\) is \(C^1\) to conclude that \(\Gamma\) is analytic.

MSC:

35R35 Free boundary problems for PDEs
35J50 Variational methods for elliptic systems
35B65 Smoothness and regularity of solutions to PDEs
35Q30 Navier-Stokes equations
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