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Variational problems with two phases and their free boundary. (English) Zbl 0844.35137
The authors study the free boundary \(\Gamma=\partial\{u>0\}=\partial\{u<0\}\) of solutions \(u\) of the variational problem \(\int_\Omega|\nabla v|^2+q(x)\cdot\lambda^2(v) dx\to \min\), where \(\Omega\subset{\mathbb{R}}^n\) is open, \(0<q<\infty\) and \(\lambda(v)=\lambda_1^2\) if \(v>0\), and \(\lambda(v)=\lambda_2^2\) if \(v<0\), for some \(\lambda_1\neq\lambda_2\). The main result is that \(\Gamma\) is \(C^1\)-smooth if \(n=2\). As in the paper by the first two authors [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)], where a similar free boundary problem is studied, they show that \(\nabla u\in L^\infty_{\text{loc}}(\Omega)\) and that the \((n-1)\)-dimensional Hausdorff measure of \(\Gamma\) is locally bounded, for arbitrary \(n\geq 2\). In contrast to the above-mentioned work, additional difficulties arise because the solutions may change sign. In order to overcome these difficulties, the authors prove a monotonicity formula which was probably inspired by a result from geometric measure theory.

MSC:
35R35 Free boundary problems for PDEs
49J40 Variational inequalities
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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