Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 6 (2017), 1317-1359.

A class of unstable free boundary problems

Serena Dipierro, Aram Karakhanyan, and Enrico Valdinoci

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We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter.

The main difference with the existing literature is that the total energy is here a nonlinear superposition of the either local or nonlocal surface tension effect with the elastic energy.

In sharp contrast with the linear case, the problem considered in this paper is unstable; namely a minimizer in a given domain is not necessarily a minimizer in a smaller domain.

We provide an explicit example for this instability. We also give a free boundary condition, which emphasizes the role played by the domain in the geometry of the free boundary. In addition, we provide density estimates for the free boundary and regularity results for the minimal solution.

As far as we know, this is the first case in which a nonlinear function of the perimeter is studied in this type of problem. Also, the results obtained in this nonlinear setting are new even in the case of the local perimeter, and indeed the instability feature is not a consequence of the possible nonlocality of the problem, but it is due to the nonlinear character of the energy functional.

Article information

Anal. PDE, Volume 10, Number 6 (2017), 1317-1359.

Received: 11 December 2015
Accepted: 9 May 2017
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R35: Free boundary problems

free boundary problems regularity nonlinear phenomena


Dipierro, Serena; Karakhanyan, Aram; Valdinoci, Enrico. A class of unstable free boundary problems. Anal. PDE 10 (2017), no. 6, 1317--1359. doi:10.2140/apde.2017.10.1317.

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  • N. Abatangelo and E. Valdinoci, “A notion of nonlocal curvature”, Numer. Funct. Anal. Optim. 35:7-9 (2014), 793–815.
  • H. W. Alt and L. A. Caffarelli, “Existence and regularity for a minimum problem with free boundary”, J. Reine Angew. Math. 325 (1981), 105–144.
  • L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, New York, 2000.
  • L. Ambrosio, G. De Philippis, and L. Martinazzi, “Gamma-convergence of nonlocal perimeter functionals”, Manuscripta Math. 134:3-4 (2011), 377–403.
  • I. Athanasopoulos, L. A. Caffarelli, C. Kenig, and S. Salsa, “An area-Dirichlet integral minimization problem”, Comm. Pure Appl. Math. 54:4 (2001), 479–499.
  • J. Bourgain, H. Brezis, and P. Mironescu, “Another look at Sobolev spaces”, pp. 439–455 in Optimal control and partial differential equations, edited by J. L. Menaldi et al., IOS, Amsterdam, 2001.
  • L. Caffarelli and E. Valdinoci, “Uniform estimates and limiting arguments for nonlocal minimal surfaces”, Calc. Var. Partial Differential Equations 41:1-2 (2011), 203–240.
  • L. A. Caffarelli, A. L. Karakhanyan, and F.-H. Lin, “The geometry of solutions to a segregation problem for nondivergence systems”, J. Fixed Point Theory Appl. 5:2 (2009), 319–351.
  • L. Caffarelli, J.-M. Roquejoffre, and O. Savin, “Nonlocal minimal surfaces”, Comm. Pure Appl. Math. 63:9 (2010), 1111–1144.
  • L. Caffarelli, O. Savin, and E. Valdinoci, “Minimization of a fractional perimeter-Dirichlet integral functional”, Ann. Inst. H. Poincaré Anal. Non Linéaire 32:4 (2015), 901–924.
  • J. Dávila, “On an open question about functions of bounded variation”, Calc. Var. Partial Differential Equations 15:4 (2002), 519–527.
  • A. Di Castro, M. Novaga, B. Ruffini, and E. Valdinoci, “Nonlocal quantitative isoperimetric inequalities”, Calc. Var. Partial Differential Equations 54:3 (2015), 2421–2464.
  • E. Di Nezza, G. Palatucci, and E. Valdinoci, “Hitchhiker's guide to the fractional Sobolev spaces”, Bull. Sci. Math. 136:5 (2012), 521–573.
  • S. Dipierro and E. Valdinoci, “On a fractional harmonic replacement”, Discrete Contin. Dyn. Syst. 35:8 (2015), 3377–3392.
  • S. Dipierro and E. Valdinoci, “Continuity and density results for a one-phase nonlocal free boundary problem”, Ann. Inst. H. Poincaré Anal. Non Linéaire (online publication November 2016).
  • S. Dipierro, A. Figalli, G. Palatucci, and E. Valdinoci, “Asymptotics of the $s$-perimeter as $s\searrow0$”, Discrete Contin. Dyn. Syst. 33:7 (2013), 2777–2790.
  • S. Dipierro, A. Figalli, and E. Valdinoci, “Strongly nonlocal dislocation dynamics in crystals”, Comm. Partial Differential Equations 39:12 (2014), 2351–2387.
  • S. Dipierro, O. Savin, and E. Valdinoci, “Boundary behavior of nonlocal minimal surfaces”, J. Funct. Anal. 272:5 (2017), 1791–1851.
  • L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, Boca Raton, FL, 1992.
  • A. Figalli, N. Fusco, F. Maggi, V. Millot, and M. Morini, “Isoperimetry and stability properties of balls with respect to nonlocal energies”, Comm. Math. Phys. 336:1 (2015), 441–507.
  • R. L. Frank, E. H. Lieb, and R. Seiringer, “Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators”, J. Amer. Math. Soc. 21:4 (2008), 925–950.
  • N. Garofalo and F.-H. Lin, “Monotonicity properties of variational integrals, $A_p$ weights and unique continuation”, Indiana Univ. Math. J. 35:2 (1986), 245–268.
  • M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105, Princeton University Press, 1983.
  • C. Imbert, “Level set approach for fractional mean curvature flows”, Interfaces Free Bound. 11:1 (2009), 153–176.
  • F. Maggi, Sets of finite perimeter and geometric variational problems: an introduction to geometric measure theory, Cambridge Studies in Advanced Mathematics 135, Cambridge University Press, 2012.
  • V. Maz'ya and T. Shaposhnikova, “On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces”, J. Funct. Anal. 195:2 (2002), 230–238.
  • O. Savin and E. Valdinoci, “Regularity of nonlocal minimal cones in dimension 2”, Calc. Var. Partial Differential Equations 48:1-2 (2013), 33–39.
  • O. Savin and E. Valdinoci, “Density estimates for a variational model driven by the Gagliardo norm”, J. Math. Pures Appl. $(9)$ 101:1 (2014), 1–26.
  • J. Yeh, Real analysis: theory of measure and integration, 2nd ed., World Scientific, Hackensack, NJ, 2006.