A limiting free boundary problem with gradient constraint and Tug-of-War games

  • P. Blanc
  • J. V. da Silva
  • J. D. RossiEmail author


In this manuscript we deal with regularity issues and the asymptotic behaviour (as \(p \rightarrow \infty \)) of solutions for elliptic free boundary problems of \(p-\)Laplacian type (\(2 \le p< \infty \)):
$$\begin{aligned} -\Delta _p u(x) + \lambda _0(x)\chi _{\{u>0\}}(x) = 0 \quad \text{ in } \quad \Omega \subset {\mathbb {R}}^N, \end{aligned}$$
with a prescribed Dirichlet boundary data, where \(\lambda _0>0\) is a bounded function and \(\Omega \) is a regular domain. First, we prove the convergence as \(p\rightarrow \infty \) of any family of solutions \((u_p)_{p\ge 2}\), as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation,
$$\begin{aligned} \left\{ \begin{array}{lllll} \max \left\{ -\Delta _{\infty } u_{\infty }, \,\, -|\nabla u_{\infty }| + \chi _{\{u_{\infty }>0\}}\right\} &{} = &{} 0 &{} \text{ in } &{} \Omega \cap \{u_{\infty } \ge 0\} \\ u_{\infty } &{} = &{} F &{} \text{ on } &{} \partial \Omega . \end{array} \right. \end{aligned}$$
Next, we obtain uniqueness for solutions to this limit problem. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game.


Lipschitz regularity estimates Free boundary problems \(\infty \)-Laplace operator Existence/uniqueness of solutions Tug-of-War games 

Mathematics Subject Classification

35J92 35D40 91A80 



This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina).


  1. 1.
    Alt, H.W., Phillips, D.: A free boundary problem for semilinear elliptic equations. J. Reine Angew. Math. 368, 63–107 (1986)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. (N.S.) 41(4), 439–505 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blanc, P., Pinasco, J.P., Rossi, J.D.: Maximal operators for the \(p-\)Laplacian family. Pac. J. Math. 287(2), 257–295 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Crandall, M.G., Evans, L.C., Gariepy, R.F.: Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Part. Differ. Equ. 13(2), 123–139 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Crandall, M.G., Gunnarsson, G., Wang, P.Y.: Uniqueness of \(\infty -\)harmonic functions and the eikonal equation. Comm. Part. Differ. Equ. 32(10–12), 1587–1615 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    da Silva, J.V., Rossi, J.D., Salort, A.: Maximal solutions for the \(\infty -\)eigenvalue problem. Adv. Calc. Var. (to appear)
  8. 8.
    Díaz, J.I.: Nonlinear Partial Differential Equations and Free Boundaries Vol. I. Elliptic equations. Research Notes in Mathematics, 106. Pitman (Advanced Publishing Program), Boston, MA, 1985. vii+323 pp. ISBN: 0-273-08572-7Google Scholar
  9. 9.
    Evans, L.C., Smart, C.K.: Everywhere differentiability of infinity harmonic functions. Calc. Var. Part. Differ. Equ. 42(1–2), 289–299 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jensen, R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123(1), 51–74 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Juutinen, P., Lindqvist, P., Manfredi, J.J.: The \(\infty \)-eigenvalue problem. Arch. Ration. Mech. Anal. 148(2), 89–105 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Juutinen, P., Parviainen, M., Rossi, J.D.: Discontinuous gradient constraints and the infinity Laplacian. Int. Math. Res. Not. IMRN 8, 2451–2492 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karp, L., Kilpeläinen, T., Petrosyan, A., Shahgholian, H.: On the porosity of free boundaries in degenerate variational inequalities. J. Differ. Equ. 164(1), 110–117 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Koskela, P., Rohde, S.: Hausdorff dimension and mean porosity. Math. Ann. 309(4), 593–609 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lee, K.-A., Shahgholian, H.: Hausdorff measure and stability for the \(p\)-obstacle problem \((2 < p < \infty )\). J. Differ. Equ. 195(1), 14–24 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lindqvist, P., Lukkari, T.: A curious equation involving the \(\infty \)-Laplacian. Adv. Calc. Var. 3(4), 409–421 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Manfredi, J.J., Parviainen, M., Rossi, J.D.: An asymptotic mean value characterization for \(p-\)harmonic functions. Proc. Am. Math. Soc. 138(3), 881–889 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Manfredi, J.J., Parviainen, M., Rossi, J.D.: On the definition and properties of \(p\)-harmonious functions. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(2), 215–241 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Manfredi, J.J.: Regularity for minima of functionals with \(p\)-growth. J. Differ. Equ. 76(2), 203–212 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Peres, Y., Schramm, O., Sheffield, S., Wilson, D.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167–210 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rossi, J.D., Teixeira, E.V.: A limiting free boundary problem ruled by Aronsson’s equation. Trans. Am. Math. Soc. 364(2), 703–719 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rossi, J.D., Teixeira, E.V., Urbano, J.M.: Optimal regularity at the free boundary for the infinity obstacle problem. Interfaces Free Bound 17(3), 381–398 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rossi, J.D., Wang, P.: The limit as \(p \rightarrow \infty \) in a two-phase free boundary problem for the \(p\)-Laplacian. Interfaces Free Bound. 18(1), 115–135 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.FCEyN, Department of MathematicsUniversidad de Buenos AiresBuenos AiresArgentina

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