manuscripta mathematica

, Volume 42, Issue 2–3, pp 199–209 | Cite as

Coincidence set of minimal surfaces for the thin obstacle

  • Ioannis Athanasopoulos
Article

Abstract

We consider the Thin Obstacle Problem for minimal surfaces in two dimensions. The coincidence set for an analytic obstacle is proved to be a finite union of intervals. We show also that the topological structure of the coincidence set is generically identical to the above in the space of twice-continuously differentiable obstacles.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I. ATHANASOPOULOS. Stability of the Coincidence set for the Signoini problem. Indiana Univ. Math. J.,30, 235–248, (1981)Google Scholar
  2. [2]
    E.F. BECHENBACH-T. RADO. Subharmonic Functions and Mineral Surfaces. Trans. Amer. Math. Soc.,35, 648–661, (1933)Google Scholar
  3. [3]
    J. FREHSE. On Signini Problem and Variational Problems with Thin Obstacles. Ann. Scuola Norm Sup. Pisa, Sec. IV, vol.IV, 343–362, (1977)Google Scholar
  4. [4]
    D. GILBARG-N.S. TRUDINGER. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, (1977)Google Scholar
  5. [5]
    S. HILDERRANDT-J.C.C. NITSCHE. Optinal Boundary Regularity for Minimal Surfaces with a Free Boundary, Manuscript Math.33, 357–364, (1981)Google Scholar
  6. [6]
    D. KINDERLEHRER. Minimal Surface whose Boundaries Contain Spikes. J. Math and Mech.19.9, 829–852, (1970)Google Scholar
  7. [7]
    D. KINDERLEHRER. The Regularity of Minimal Surface Defined over Slit Domains. Pacific Journal of Mathematics,37.1, 109–117, (1971)Google Scholar
  8. [8]
    D. KINDERLEHRER. The smoothness of the solution of the boundary obstacle problem. J. Math pures et appl.,60, 193–212, (1981)Google Scholar
  9. [9]
    H. LEWY. On the boundary behavior of minimal surfaces. Proc. N.A.S.,37, 103–110, (1951)Google Scholar
  10. [10]
    J.L. LIONS-G. STAMPACCHIA. Vatiational Inequalities Comm. Pure Appl. Math.20, 493–519, (1967)Google Scholar
  11. [11]
    C. MORREY, Jr. On the analyticity of the solutions of analytic non-linear elliptic systems of P.D.E. II, Analyticity at the coundary. Amer. J. Math.80, 219–237, (1958)Google Scholar
  12. [12]
    J.C.C. NITSCHE. Variational problems with inequalities as boundary conditions or how to fashion a cheap hat for Giacometti's brother. Arch. Rational Mech Anal.,35, 83–113, (1969)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Ioannis Athanasopoulos
    • 1
  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA

Personalised recommendations