Inventiones mathematicae

, Volume 187, Issue 3, pp 535–587 | Cite as

On the singularities of a free boundary through Fourier expansion

  • John Andersson
  • Henrik Shahgholian
  • Georg S. Weiss
Article

Abstract

In this paper we are concerned with singular points of solutions to the unstable free boundary problem
$$\Delta u = - \chi_{\{u>0\}} \quad\hbox{in } B_1.$$
The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics.

It is known that solutions to the above problem may exhibit singularities—that is points at which the second derivatives of the solution are unbounded—as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the nonlinearity χ {u>0}.

The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in ℝ3.

A surprising fact in ℝ3 is that although
$$\frac{u(r\mathbf{x})}{\sup_{B_1}|u(r\mathbf{x})|}$$
can converge at singularities to each of the harmonic polynomials
$$xy,\qquad {x^2+y^2\over2}-z^2\quad \textrm{and}\quad z^2-{x^2+y^2\over2},$$
it may not converge to any of the non-axially-symmetric harmonic polynomials α((1+δ)x 2+(1−δ)y 2−2z 2) with δ≠1/2.

We also prove the existence of stable singularities in ℝ3.

Mathematics Subject Classification (2000)

35R35 35B40 35J60 

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References

  1. 1.
    Ambrosetti, A., Struwe, M.: Existence of steady vortex rings in an ideal fluid. Arch. Ration. Mech. Anal. 108(2), 97–109 (1989) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Andersson, J., Shahgholian, H., Weiss, G.S.: Uniform regularity close to cross singularities in an unstable free boundary problem. Commun. Math. Phys. (2010) Google Scholar
  3. 3.
    Andersson, J., Weiss, G.S.: Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem. J. Differ. Equ. 228(2), 633–640 (2006) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Blank, I.: Eliminating mixed asymptotics in obstacle type free boundary problems. Commun. Partial Differ. Equ. 29(7–8), 1167–1186 (2004) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Caffarelli, L.A.: The regularity of free boundaries in higher dimensions. Acta Math. 139(3–4), 155–184 (1977) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4(4–5), 383–402 (1998) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Caffarelli, L.A., Rivière, N.M.: Asymptotic behaviour of free boundaries at their singular points. Ann. Math. (2) 106(2), 309–317 (1977) MATHCrossRefGoogle Scholar
  8. 8.
    Chanillo, S., Grieser, D., Imai, M., Kurata, K., Ohnishi, I.: Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun. Math. Phys. 214(2), 315–337 (2000) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chanillo, S., Grieser, D., Kurata, K.: The free boundary problem in the optimization of composite membranes. In: Differential Geometric Methods in the Control of Partial Differential Equations, Boulder, CO, 1999. Contemp. Math., vol. 268, pp. 61–81. Am. Math. Soc., Providence (2000) CrossRefGoogle Scholar
  10. 10.
    Chanillo, S., Kenig, C.E.: Weak uniqueness and partial regularity for the composite membrane problem. J. Eur. Math. Soc. 10(3), 705–737 (2008) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chanillo, S., Kenig, C.E., To, T.: Regularity of the minimizers in the composite membrane problem in ℝ2. J. Funct. Anal. 255(9), 2299–2320 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Díaz, J.I., Shmarev, S.: Lagrangian approach to the study of level sets: application to a free boundary problem in climatology. Arch. Ration. Mech. Anal. 194(1), 75–103 (2009) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ganzburg, M.I.: Polynomial inequalities on measurable sets and their applications. Constr. Approx. 17(2), 275–306 (2001) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Karp, L., Margulis, A.S.: Newtonian potential theory for unbounded sources and applications to free boundary problems. J. Anal. Math. 70, 1–63 (1996) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Monneau, R., Weiss, G.S.: An unstable elliptic free boundary problem arising in solid combustion. Duke Math. J. 136(2), 321–341 (2007) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Pacard, F.: Partial regularity for weak solutions of a nonlinear elliptic equation. Manuscr. Math. 79(2), 161–172 (1993) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Price, P.: A monotonicity formula for Yang-Mills fields. Manuscr. Math. 43(2–3), 131–166 (1983) MATHCrossRefGoogle Scholar
  18. 18.
    Rivière, T.: A lower-epiperimetric inequality for area-minimizing surfaces. Commun. Pure Appl. Math. 57(12), 1673–1685 (2004) MATHCrossRefGoogle Scholar
  19. 19.
    Rivière, T., Tian, G.: The singular set of 1-1 integral currents. Ann. Math. (2) 169(3), 741–794 (2009) MATHCrossRefGoogle Scholar
  20. 20.
    Schoen, R.M.: Analytic aspects of the harmonic map problem. In: Seminar on Nonlinear Partial Differential Equations, Berkeley, CA, 1983. Math. Sci. Res. Inst. Publ., vol. 2, pp. 321–358. Springer, New York (1984) CrossRefGoogle Scholar
  21. 21.
    Shahgholian, H.: The singular set for the composite membrane problem. Commun. Math. Phys. 271(1), 93–101 (2007) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Simon, L.: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. (2) 118(3), 525–571 (1983) MATHCrossRefGoogle Scholar
  23. 23.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, Vol. 30. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  24. 24.
    Weiss, G.S.: Partial regularity for weak solutions of an elliptic free boundary problem. Commun. Partial Differ. Equ. 23(3–4), 439–455 (1998) MATHCrossRefGoogle Scholar
  25. 25.
    White, B.: Tangent cones to two-dimensional area-minimizing integral currents are unique. Duke Math. J. 50(1), 143–160 (1983) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • John Andersson
    • 1
  • Henrik Shahgholian
    • 2
  • Georg S. Weiss
    • 3
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of MathematicsRoyal Institute of TechnologyStockholmSweden
  3. 3.Mathematical Institute of the Heinrich Heine UniversityDüsseldorfGermany

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