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On the number of singularities for the obstacle problem in two dimensions. (English) Zbl 1041.35093

The problem here considered is the classical obstacle problem for an elastic membrane in the gravity field in its simplest formulation (flat horizontal obstacle \((z=0)\), membrane hanging from a flat horizontal smooth profile \((z=\lambda)\)). The results are particularly interesting and are concerned with the possible singular points of the free boundary (i.e. the boundary of the coincidence set). A first result proves the regularity conjecture by D. G. Schaeffer (1972) in the two-dimensional case: for almost every \(\lambda =0\) the free boundary is an analytic one-dimensional submanifold of \(\mathbb{R}^2\). The next fundamental result is concerned with an upper bound of the number of possible singular points, still in \(\mathbb{R}^2\): let \({C}\) be an open connected component of the coincidence set, then there exists a constant \(\rho >0\) depending on \(| \Delta^2u| _{L^\infty (\Omega)}, d({C},\partial \Omega)\) such that
(i) if diam \(({C})<\rho\) the free boundary is analytic except in not more than two points,
(ii) if diam \(({C})\geq \rho\), then all singular points have mutual distances not less than \(\dfrac{\rho}{2}\).
Clearly the latter result also provides an upper bound for the number of singular points in terms of \(\rho\) and of \(| \Omega| \). Various generalization to more general problems are proposed. A basic tool used in the proof is a monotonicity formula which allows to use Taylor’s expansion for \(u\) near singular points. In proving his results the author makes extensive use of the now classical theory on obstacle problems developed by many of the most distinguished mathematicians of our time. A nice paper.

MSC:

35R35 Free boundary problems for PDEs
74K15 Membranes
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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