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The fictitious domain method as applied to the Signorini problem. (English. Russian original) Zbl 1161.35028
Dokl. Math. 68, No. 2, 161-166 (2003); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 392, No. 1, 17-20 (2003).
The authors deal with the Signorini problem,
\[ \begin{gathered} -\text{div}(a\nabla u)= f\quad\text{in }\Omega,\\ u= 0\quad\text{on }\Gamma_0,\\ u\geq 0,\;a{\partial u\over\partial\vec n}\leq 0,\;ua{\partial u\over\partial\vec n}= 0\quad\text{on }\Gamma_c,\end{gathered}\tag{1} \]
where \(\Omega\subset\mathbb{R}^2\) is a bounded simply connected domain with smooth boundary \(\Gamma= \Gamma_c\cup \Gamma_0\), \(\Gamma_c\cap \Gamma_0\neq\emptyset\), and \(\mathrm{mes}\Gamma_0> 0\); \(\vec n\) is the inward normal vector with respect to \(\Gamma\). The coefficient \(a\in L^\infty_{\text{loc}}(\mathbb{R}^2)\) and \(f\in L^\infty_{\text{loc}}(\mathbb{R}^2)\) in (1) are given functions, \(a\geq \gamma > 0\), for some constant \(\gamma\). To prove the solvability of (1) the authors apply the fictitious domain method, which involves the construction of a family of auxiliary problems defined in a wider domain and possessing the property that the limit of their solutions converges to the solution of the original problem.
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