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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

Citations:

Zbl 0928.35213
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References:

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