Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 2 (2016), 487-502.

Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions

Emanuel Indrei and Andreas Minne

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Abstract

In the study of classical obstacle problems, it is well known that in many configurations, the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper, we employ a different approach and prove tangential touch of free and fixed boundaries in two dimensions for fully nonlinear elliptic operators. Along the way, several n-dimensional results of independent interest are obtained, such as BMO-estimates, C1,1-regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.

Article information

Source
Anal. PDE, Volume 9, Number 2 (2016), 487-502.

Dates
Received: 12 June 2015
Revised: 6 January 2016
Accepted: 9 February 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843244

Digital Object Identifier
doi:10.2140/apde.2016.9.487

Mathematical Reviews number (MathSciNet)
MR3513142

Zentralblatt MATH identifier
1341.35054

Subjects
Primary: 35JXX 35QXX
Secondary: 49SXX

Keywords
obstacle problem tangential touch fully nonlinear equations nontransverse intersection free boundary problem

Citation

Indrei, Emanuel; Minne, Andreas. Nontransversal intersection of free and fixed boundaries for fully nonlinear elliptic operators in two dimensions. Anal. PDE 9 (2016), no. 2, 487--502. doi:10.2140/apde.2016.9.487. https://projecteuclid.org/euclid.apde/1510843244


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References

  • H. W. Alt and G. Gilardi, “The behavior of the free boundary for the dam problem”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 9:4 (1982), 571–626.
  • J. Andersson, “On the regularity of a free boundary near contact points with a fixed boundary”, J. Differential Equations 232:1 (2007), 285–302.
  • J. Andersson, N. Matevosyan, and H. Mikayelyan, “On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem”, Ark. Mat. 44:1 (2006), 1–15.
  • J. Andersson, E. Lindgren, and H. Shahgholian, “Optimal regularity for the no-sign obstacle problem”, Comm. Pure Appl. Math. 66:2 (2013), 245–262.
  • D. E. Apushkinskaya and N. N. Uraltseva, “On the behavior of the free boundary near the boundary of the domain”, Zap. Nauchn. Sem. POMI 221 (1995), 5–19. In Russian; translated in J. Math. Sci. New York 87:2 (1997), 3267–3276.
  • L. A. Caffarelli and X. Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995.
  • L. A. Caffarelli and G. Gilardi, “Monotonicity of the free boundary in the two-dimensional dam problem”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 7:3 (1980), 523–537.
  • L. Caffarelli, M. G. Crandall, M. Kocan, and A. Swi\polhkech, “On viscosity solutions of fully nonlinear equations with measurable ingredients”, Comm. Pure Appl. Math. 49:4 (1996), 365–397.
  • A. Figalli and H. Shahgholian, “A general class of free boundary problems for fully nonlinear elliptic equations”, Arch. Ration. Mech. Anal. 213:1 (2014), 269–286.
  • E. Indrei and A. Minne, “Regularity of solutions to fully nonlinear elliptic and parabolic free boundary problems”, Ann. Inst. H. Poincaré Anal. Non Linéaire (online publication May 11 2015).
  • N. V. Krylov, “Boundedly inhomogeneous elliptic and parabolic equations”, Izv. Akad. Nauk SSSR Ser. Mat. 46:3 (1982), 487–523, 670. In Russian; translated in Math. USSR Izv. 20:3 (1983), 459–492.
  • N. Matevosyan, “Tangential touch between free and fixed boundaries in a problem from superconductivity”, Comm. Partial Differential Equations 30:7-9 (2005), 1205–1216.
  • N. Matevosyan and P. A. Markowich, “Behavior of the free boundary near contact points with the fixed boundary for nonlinear elliptic equations”, Monatsh. Math. 142:1-2 (2004), 17–25.
  • A. Petrosyan, H. Shahgholian, and N. Uraltseva, Regularity of free boundaries in obstacle-type problems, Graduate Studies in Mathematics 136, American Mathematical Society, Providence, RI, 2012.
  • M. Safonov, “On the boundary value problems for fully nonlinear elliptic equations of second order”, Mathematics Research Report No. MRR 049-94, The Australian National University, Canberra, 1994, hook http://www.math.umn.edu/~safon002/NOTES/BVP_94/BVP.pdf \posturlhook.
  • H. Shahgholian and N. Uraltseva, “Regularity properties of a free boundary near contact points with the fixed boundary”, Duke Math. J. 116:1 (2003), 1–34.
  • L. Wang, “On the regularity theory of fully nonlinear parabolic equations, II”, Comm. Pure Appl. Math. 45:2 (1992), 141–178.
  • N. Winter, “$W\sp {2,p}$ and $W\sp {1,p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations”, Z. Anal. Anwend. 28:2 (2009), 129–164.