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.

MathOverflow Questions:

Gradient estimates for a boundary value problem

MSC:

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

Zbl 0449.35105
Full Text:

References:

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