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Zbl 0823.35068
Sakaguchi, Shigeru
Critical points of solutions to the obstacle problem in the plane. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 21, No.2, 157-173 (1994). ISSN 0391-173X

The author considers the obstacle problem in a bounded simply connected domain $\Omega \subset \bbfR\sp 2$: find $u\in K$ satisfying $\int\sb \Omega \langle A(\nabla u), \nabla\nu- \nabla u\rangle dx\geq 0$ for all $\nu\in K$, where $K:= \{\nu\in H\sb 0\sp 1 (\Omega)\mid \nu(x)\geq \psi(x)$ in $\Omega\}$ and the function $\psi$ is given. The estimates of the number of critical (or saddle) points of the solution $u$ are proved. The typical result is: \par Theorem 5. Suppose that the number of connected components of local maximum points of $\psi$ is equal to $N$. Then the number of saddle points of $u$ in the noncoincidence set is finite and $\sum\sb{j=1}\sp k m\sb j \leq N$, where $m\sb 1, \dots, m\sb k$ denote the multiplicities of these points.
[U.Raitums (Riga)]

MSC 2000:: *35J85 Unilateral problems; variational inequalities (elliptic type)
35B05 General behavior of solutions of PDE

Keywords: obstacle problem; number of saddle points of the solution

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