## The behavior of the free boundary near the fixed boundary for a minimization problem.(English)Zbl 1111.35135

The paper analyzes the behaviour of the free boundary arising from the minimization problem for the functional $\int_\Omega | \nabla u(x)| ^2 +q^2(x) \lambda^2(u)$ where $$q(x)$$ is a given smooth function, $$q(x)\neq0$$ $$\forall x,$$ and $\lambda^2(u)=\begin{cases} \lambda^2_+ & \text{if}\;u>0,\cr \lambda^2_- & \text{if}\;u<0, \end{cases}$ with $$\lambda^2_+-\lambda^2_-\neq0,$$ $$\lambda_+>0,$$ $$\lambda_->0.$$
Taking $$\Omega$$ to be smooth domain and the admissible class of minimizers $$\{u\colon\;u-f\in W^{1,2}_0(\Omega)\}$$ with smooth enough $$f,$$ the main result of the paper asserts that at contact point between the fixed and the free boundary where $$f$$ and $$\nabla f$$ vanish simultaneously, the free boundary approaches the fixed one in a tangential fashion. This complements the result due to A. Gurevich [Commun. Pure Appl. Math. 52, No. 3, 363-403 (1999; Zbl 0928.35213)] which shows that if $$\nabla f=0$$ whenever $$f=0,$$ then $$f\in \text{Lip\,}(\overline\Omega).$$

### MSC:

 35R35 Free boundary problems for PDEs 49Q10 Optimization of shapes other than minimal surfaces 35B65 Smoothness and regularity of solutions to PDEs 76S05 Flows in porous media; filtration; seepage 74M15 Contact in solid mechanics

Zbl 0928.35213
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### References:

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