Abstract
We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Rivière: we state that every connected component of the interior of the coincidence set has at most N 0 singular points, where N 0 is only dependent on some geometric constants. Moreover, if the component is small enough, then this component has at most two singular points. On the other hand, we prove in a simple case a conjecture of D.G. Schaeffer on the generic regularity of the free boundary: for a family of obstacle problems in two dimensions continuously indexed by a parameter λ, the free boundary of the solution uλ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points.
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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).
Beckner, W., Kenig, C., and Pipher, J., in preparation.
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).
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).
Brezis, H.Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, (1993).
Brezis, H. and Kinderlehrer, D. The smoothness of solutions to nonlinear variational inequalities,Indiana Univ. Math. J.,23(9), 831–844, (1974).
Caffarelli, L.A. Free boundary problem in higher dimensions,Acta Math.,139, 155–184, (1977).
Caffarelli, L.A. Compactness methods in free boundary problems,Comm. Partial Differential Equations,5(4), 427–448, (1980).
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).
Caffarelli, L.A. A Harnack inequality approach to the regularity of free boundaries, Part I: Lipschitz free boundaries areC 1,α,Rev. Mat. Iberoamericana,3(2), 139–162, (1987).
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).
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).
Caffarelli, L.A. The obstacle problem revisited,J. Fourier Anal. Appl.,4(4–5), 383–402, (1998).
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).
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).
Caffarelli, L.A. and Rivière, N.M. Asymptotic behavior of free boundaries at their singular points,Ann. of Math. 106, 309–317, (1977).
Federer, H.Geometric Measure Theory, Springer-Verlag, (1969).
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).
Friedman, A.Variational Principles and Free Boundary Problems, Pure and applied mathematics, ISSN 0079-8185, a Wiley-Interscience publication, (1982).
Gilbarg, D.N. and Trudinger, N.S.Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 2nd ed., (1983).
Isakov, V. Inverse theorems on the smoothness of potentials,Differential Equations,11, 50–57, (1976).
Kato, T. Schrödinger operators with singular potentials,Israel J. Math.,13, 135–148, (1972).
Kinderlehrer, D. and Nirenberg, L. Regularity in free boundary problems,Ann. Scuola Norm. Sup. Pisa,4, 373–391, (1977).
Kinderlehrer, D. and Stampacchia, G.An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, (1980).
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).
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).
Morrey, C.B.Multiple Integrals in the Calculus of Variations, Die Grundlehrender Mathematischen Wissenschaften in Einzeldarstellungen,130, Springer-Verlag, NY, (1966).
Rodrigues, J.F.Obstacle Problems in Mathematical Physics, North-Holland, (1987).
Schaeffer, D.G. An example of generic regularity for a non-linear elliptic equation,Arch. Rat. Mach. Anal.,57, 134–141, (1974).
Schaeffer, D.G. A Stability theorem for the obstacle problem,Advances in Math.,16, 34–47, (1975).
Schaeffer, D.G. Some examples of singularities in a free boundary,Ann. Scuola Norm. Sup. Pisa,4(4), 131–144, (1976).
Schaeffer, D.G. One-sided estimates for the curvature of the free boundary in the obstacle problem,Adv. in Math.,24, 78–98, (1977).
Weiss, G.S. A homogeneity improvement approach to the obstacle problem,Invent. Math.,138, 23–50, (1999).
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Communicated by David Jerison
Dedicated to Henri Berestycki and Alexis Bonnet.
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Monneau, R. On the number of singularities for the obstacle problem in two dimensions. J Geom Anal 13, 359–389 (2003). https://doi.org/10.1007/BF02930701
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DOI: https://doi.org/10.1007/BF02930701