Arkiv för Matematik

Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions

Mahmoudreza Bazarganzadeh and Erik Lindgren

Full-text: Open access


We study minimizers of the functional $$\int_{B^+_1}(|\nabla u|^2 + 2(\lambda^+(u^+)^p+\lambda^-(u^-)^p))dx, $$where $B_{1}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .17ex\hbox {$\scriptstyle +$}} {\raise .1ex\hbox {$\scriptscriptstyle +$}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\}$, u=0 on {xB1: x1=0}, $\lambda^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .17ex\hbox {$\scriptstyle \pm $}} {\raise .1ex\hbox {$\scriptscriptstyle \pm $}} {\scriptscriptstyle \pm }}}$ are two positive constants and 0< p<1. In two dimensions, we prove that the free boundary is a uniform C1 graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {xu(x)>0}.

Article information

Ark. Mat., Volume 52, Number 1 (2014), 21-42.

Received: 8 February 2012
Revised: 10 September 2012
First available in Project Euclid: 30 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2013 © Institut Mittag-Leffler


Bazarganzadeh, Mahmoudreza; Lindgren, Erik. Tangential touch between the free and the fixed boundary in a semilinear free boundary problem in two dimensions. Ark. Mat. 52 (2014), no. 1, 21--42. doi:10.1007/s11512-012-0179-3.

Export citation


  • Alt, H. W., Caffarelli, L. and Friedman, A., Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), 431–461.
  • Alt, H. W. and Phillips, D., A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math. 368 (1986), 63–107.
  • Andersson, J., Matevosyan, N. and Mikayelyan, H., On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem, Ark. Mat. 44 (2006), 1–15.
  • Caffarelli, L. A., Karp, L. and Shahgholian, H., Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math. 151 (2000), 269–292.
  • Caffarelli, L. A. and Rivière, N. M., Smoothness and analyticity of free boundaries in variational inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. 3 (1976), 289–310.
  • Campanato, S., Proprietà di hölderianità di alcune classi di funzioni, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (1963), 175–188.
  • Evans, L. C., Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics 19, Amer. Math. Soc., Providence, RI, 2010.
  • Giaquinta, M. and Giusti, E., Sharp estimates for the derivatives of local minima of variational integrals, Boll. Unione Mat. Ital. Sez. A Mat. Soc. Cult. 3 (1984), 239–248.
  • Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, Berlin, 1998.
  • Karakhanyan, A. L., Kenig, C. E. and Shahgholian, H., The behavior of the free boundary near the fixed boundary for a minimization problem, Calc. Var. Partial Differential Equations 28 (2007), 15–31.
  • Lindgren, E. and Petrosyan, A., Regularity of the free boundary in a two-phase semilinear problem in two dimensions, Indiana Univ. Math. J. 57 (2008), 3397–3417.
  • Lindgren, E. and Silvestre, L., On the regularity of a singular variational problem. Preprint, 2005.
  • Phillips, D., Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Partial Differential Equations 8 (1983), 1409–1454.
  • Phillips, D., A minimization problem and the regularity of solutions in the presence of a free boundary, Indiana Univ. Math. J. 32 (1983), 1–17.
  • Shahgholian, H., C1,1 regularity in semilinear elliptic problems, Comm. Pure Appl. Math. 56 (2003), 278–281.
  • Shahgholian, H., Uraltseva, N. N. and Weiss, G. S., The two-phase membrane problem—regularity of the free boundaries in higher dimensions, Int. Math. Res. Not. IMRN 2007 (2007), No. 8, Art. ID rnm026.
  • Uraltseva, N. N., Two-phase obstacle problem, J. Math. Sci. (N. Y.) 106 (2001), 3073–3077.
  • Weiss, G. S., Partial regularity for weak solutions of an elliptic free boundary problem, Comm. Partial Differential Equations 23 (1998), 439–455.
  • Weiss, G. S., An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary, Interfaces Free Bound. 3 (2001), 121–128.
  • Weiss, G. S., Boundary monotonicity formulae and applications to free boundary problems. I. The elliptic case, Electron. J. Differential Equations 2004 (2004), no. 44.