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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References:

 [1] Alt, H.W., Caffarelli, L.A., and Friedman, A. Variational Problems with two phases and their free boundaries,Trans. Am. Math. Soc.,282(2), 431–461, (1984). · Zbl 0844.35137 [2] Beckner, W., Kenig, C., and Pipher, J., in preparation. [3] Berestycki, H. and Nirenberg, L. On the method of moving planes and the sliding method,Bol. Soc. Brasil. Mat. (N.S.),22, 1–37, (1991). · Zbl 0784.35025 [4] Blank, I. Sharp results for the regularity and stability of the free boundary in the obstacle problem,Indiana Univ. Math. J.,50(3), 1077–1112, (2002). · Zbl 1032.35170 [5] Brezis, H.Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, (1993). [6] Brezis, H. and Kinderlehrer, D. The smoothness of solutions to nonlinear variational inequalities,Indiana Univ. Math. J.,23(9), 831–844, (1974). · Zbl 0278.49011 [7] Caffarelli, L.A. Free boundary problem in higher dimensions,Acta Math.,139, 155–184, (1977). · Zbl 0386.35046 [8] Caffarelli, L.A. Compactness methods in free boundary problems,Comm. Partial Differential Equations,5(4), 427–448, (1980). · Zbl 0437.35070 [9] Caffarelli, L.A. A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets,Boll. Un. Mat. Ital. A,18(5), 109–113, (1981). · Zbl 0453.35085 [10] Caffarelli, L.A. A Harnack inequality approach to the regularity of free boundaries, Part I: Lipschitz free boundaries areC 1,{$$\alpha$$},Rev. Mat. Iberoamericana,3(2), 139–162, (1987). · Zbl 0676.35085 [11] Caffarelli, L.A. A Harnack inequality approach to the regularity of free boundaries, Part II: Flat free boundaries are Lipschitz,Comm. Pure Appl. Math.,42, 55–78, (1989). · Zbl 0676.35086 [12] Caffarelli, L.A. A Harnack inequality approach to the regularity of free boundaries, Part III: existence theory, compactness, and dependence on X,Ann. Scu. Norm. Sup. Pisa, Cl. Sci.,15(4), 583–602, (1989). · Zbl 0702.35249 [13] Caffarelli, L.A. The obstacle problem revisited,J. Fourier Anal. Appl.,4(4–5), 383–402, (1998). · Zbl 0928.49030 [14] Caffarelli, L.A., Kenig, C.E., and Jerison, D. Some new monotonicity theorems with applications to free boundary problems,Ann. Math. (2),155(2), 369–404, (2002). · Zbl 1142.35382 [15] Caffarelli, L.A. and Rivière, N.M. Smoothness and analyticity of free boundaries in variational inequalities,Ann. Scuola Norm. Sup. Pisa,3(4), 289–310, (1975). · Zbl 0363.35009 [16] Caffarelli, L.A. and Rivière, N.M. Asymptotic behavior of free boundaries at their singular points,Ann. of Math. 106, 309–317, (1977). · Zbl 0364.35041 [17] Federer, H.Geometric Measure Theory, Springer-Verlag, (1969). · Zbl 0176.00801 [18] Frehse, J. On the regularity of the solution of a second order variational inequality,Boll. Un. Mat. Ital. B (7),6(4), 312–315, (1972). · Zbl 0261.49021 [19] Friedman, A.Variational Principles and Free Boundary Problems, Pure and applied mathematics, ISSN 0079-8185, a Wiley-Interscience publication, (1982). · Zbl 0564.49002 [20] Gilbarg, D.N. and Trudinger, N.S.Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2nd ed., (1983). · Zbl 0562.35001 [21] Isakov, V. Inverse theorems on the smoothness of potentials,Differential Equations,11, 50–57, (1976). · Zbl 0328.31010 [22] Kato, T. Schrödinger operators with singular potentials,Israel J. Math.,13, 135–148, (1972). · Zbl 0246.35025 [23] Kinderlehrer, D. and Nirenberg, L. Regularity in free boundary problems,Ann. Scuola Norm. Sup. Pisa,4, 373–391, (1977). · Zbl 0352.35023 [24] Kinderlehrer, D. and Stampacchia, G.An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, (1980). · Zbl 0457.35001 [25] Monneau, R. Problèmes de frontières libres, EDP elliptiques non linéaires et application en combustion, supra-conductivité et élasticité, Doctoral Dissertation, Université Pierre et Marie Curie, Paris, (1999). [26] Monneau, R. A brief overview on the obstacle problem inProceedings of the Third European Congress of Mathematics, Barcelona, (2000): Progress in Mathematics,202, Birkhäuser Verlag, Basel/Switzerland, 303–312, (2001). [27] Morrey, C.B.Multiple Integrals in the Calculus of Variations, Die Grundlehrender Mathematischen Wissenschaften in Einzeldarstellungen,130, Springer-Verlag, NY, (1966). · Zbl 0142.38701 [28] Rodrigues, J.F.Obstacle Problems in Mathematical Physics, North-Holland, (1987). · Zbl 0606.73017 [29] Schaeffer, D.G. An example of generic regularity for a non-linear elliptic equation,Arch. Rat. Mach. Anal.,57, 134–141, (1974). · Zbl 0319.35036 [30] Schaeffer, D.G. A Stability theorem for the obstacle problem,Advances in Math.,16, 34–47, (1975). · Zbl 0317.49013 [31] Schaeffer, D.G. Some examples of singularities in a free boundary,Ann. Scuola Norm. Sup. Pisa,4(4), 131–144, (1976). [32] Schaeffer, D.G. One-sided estimates for the curvature of the free boundary in the obstacle problem,Adv. in Math.,24, 78–98, (1977). · Zbl 0354.35075 [33] Weiss, G.S. A homogeneity improvement approach to the obstacle problem,Invent. Math.,138, 23–50, (1999). · Zbl 0940.35102
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