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On the Dirichlet and Serrin Problems for the Inhomogeneous Infinity Laplacian in Convex Domains: Regularity and Geometric Results

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Abstract

Given an open bounded subset Ω of \({\mathbb{R}^n}\), which is convex and satisfies an interior sphere condition, we consider the pde \({-\Delta_{\infty} u = 1}\) in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class C 1(Ω). We then investigate the overdetermined Serrin-type problem, formerly considered in Buttazzo and Kawohl (Int Math Res Not, pp 237–247, 2011), obtained by adding the extra boundary condition \({|\nabla u| = a}\) on ∂Ω; by using a suitable P-function we prove that, if Ω satisfies the same assumptions as above and in addition contains a ball which touches ∂Ω at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of Ω coincide. In turn, in dimension n = 2, this entails that Ω must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class C 2.

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References

  1. Alvarez, O., Lasry, J., Lions, P.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265–288 (1997). doi:10.1016/S0021-7824(97)89952-7

  2. Ambrosio, L., Crippa, G.: Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Transport Equations and Multi-D Hyperbolic Conservation Laws. Lect. Notes Unione Mat. Ital., vol. 5, pp. 3–57. Springer, Berlin, 2008

  3. Aronsson G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6, 551–561 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aronsson, G., Crandall, M., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. (N.S.) 41(4), 439–505 (2004)

  5. Barles G.: Uniqueness and regularity results for first-order Hamilton–Jacobi equations. Indiana Univ. Math. J. 39(2), 443–466 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barron E.N., Jensen R.R., Wang C.Y.: The Euler equation and absolute minimizers of L ∞ functionals. Arch. Ration. Mech. Anal. 157(4), 255–283 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bhattacharya T., Mohammed A.: Inhomogeneous Dirichlet problems involving the infinity-Laplacian. Adv. Differ. Equ. 17(3–4), 225–266 (2012)

    MATH  MathSciNet  Google Scholar 

  8. Brock, F., Henrot, A.: A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative. Rend. Circ. Mat. Palermo (2) 51(3): 375–390 (2002)

  9. Brock F., Prajapat J.: Some new symmetry results for elliptic problems on the sphere and in Euclidean space. Rend. Circ. Mat. Palermo 49, 445–462 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Buttazzo G., Kawohl B.: Overdetermined boundary value problems for the ∞-Laplacian. Int. Math. Res. Not. 2, 237–247 (2011)

    MathSciNet  Google Scholar 

  11. Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, vol. 58. Birkhäuser, Boston, 2004

  12. Cannarsa P., Yu Y.: Singular dynamics for semiconcave functions. J. Eur. Math. Soc. 11(5), 999–1024 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Crandall M., Evans L., Gariepy R.: Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differ. Equ. 13(2), 123–139 (2001)

    MATH  MathSciNet  Google Scholar 

  14. Crandall, M., Ishii, H., Lions, P.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1–67 (1992)

  15. Crasta, G., Fragalà, I.: A symmetry problem for the infinity Laplacian. Int. Mat. Res. Not. (2014). doi:10.1093/imrn/rnu204

  16. Crasta, G., Fragalà, I.: Geometric issues in PDE problems related to the infinity Laplace operator (2015, preprint)

  17. Crasta, G., Fragalà, I.: On the characterization of some classes of proximally smooth sets. ESAIM Control Optim. Calc. Var. (2015). doi:10.1051/cocv/2015022

  18. Crasta G., Fragalà I., Gazzola F.: A sharp upper bound for the torsional rigidity of rods by means of web functions. Arch. Ration. Mech. Anal. 164, 189–211 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Evans, L., Savin, O.: C 1,α regularity for infinity harmonic functions in two dimensions. Calc. Var. Partial Differ. Equ. 32, 325–347 (2008)

  20. Evans L., Smart C.: Everywhere differentiability of infinity harmonic functions. Calc. Var. Partial Differ. Equ. 42, 289–299 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Fragalà, I.: Symmetry results for overdetermined problems on convex domains via Brunn–Minkowski inequalities. J. Math. Pures Appl. (9) 97(1), 55–65 (2012)

  22. Fragalà, I., Gazzola, F.: Partially overdetermined elliptic boundary value problems. J. Differ. Equ. 245, 1299–1322 (2008)

  23. Fragalà, I., Gazzola, F., Kawohl, B.: Overdetermined problems with possibly degenerate ellipticity, a geometric approach. Math. Z. 254, 117–132 (2006)

  24. Garofalo N., Lewis J.: A symmetry result related to some overdetermined boundary value problems. Am. J. Math. 111, 9–33 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  25. Georgiev P., Zlateva N.: Reconstruction of the Clarke subdifferential by the Lasry–Lions regularizations. J. Math. Anal. Appl. 248(2), 415–428 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hale, J.: Ordinary Differential Equations, 2nd edn. Robert E. Krieger Publishing Co. Inc., Huntington, 1980

  27. Hong G.: Boundary differentiability for inhomogeneous infinity Laplace equations. Electron. J. Differ. Equ. 72, 6 (2014)

  28. Hong, G.: Counterexample to C 1 boundary regularity of infinity harmonic functions. Nonlinear Anal. 104, 120–123 (2014)

  29. Jensen R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123, 51–74 (1993)

    Article  MATH  Google Scholar 

  30. Kawohl, B.: Overdetermined problems and the p-Laplacian. Acta Math. Univ. Comen. (N.S.) 76, 77–83 (2007)

  31. Lindgren, E.: On the regularity of solutions of the inhomogeneous infinity Laplace equation. Proc. Am. Math. Soc. 142(1), 277–288 (2014). doi:10.1090/S0002-9939-2013-12180-5

  32. Lu G., Wang P.: Inhomogeneous infinity Laplace equation. Adv. Math. 217, 1838–1868 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lu G., Wang P.: Infinity Laplace equation with non-trivial right-hand side. Electron. J. Differ. Equ. 77, 12 (2010)

    Google Scholar 

  34. Peres Y., Schramm O., Sheffield S., Wilson D.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167–210 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sakaguchi, S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14(3), 403–421 (1988). http://www.numdam.org/item?id=ASNSP_1987_4_14_3_403_0

  36. Serrin J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)

    MATH  MathSciNet  Google Scholar 

  37. Siljander, J., Wang, C., Zhou, Y.: Everywhere differentiability of viscosity solutions to a class of Aronsson’s equations (2014, preprint). arXiv:1409.6804

  38. Vogel A.: Symmetry and regularity for general regions having a solution to certain overdetermined boundary value problems. Atti Sem. Mat. Fis. Univ. Modena 40, 443–484 (1992)

    MATH  MathSciNet  Google Scholar 

  39. Wang C., Yu Y.: C 1-boundary regularity of planar infinity harmonic functions. Math. Res. Lett. 19(4), 823–835 (2012)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Univ. di Roma I, P.le A. Moro 2, 00185, Rome, Italy

    Graziano Crasta

  2. Dipartimento di Matematica, Politecnico, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy

    Ilaria Fragalà

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  1. Graziano Crasta
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  2. Ilaria Fragalà
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Correspondence to Graziano Crasta.

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Communicated by D. Kinderlehrer

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Crasta, G., Fragalà, I. On the Dirichlet and Serrin Problems for the Inhomogeneous Infinity Laplacian in Convex Domains: Regularity and Geometric Results. Arch Rational Mech Anal 218, 1577–1607 (2015). https://doi.org/10.1007/s00205-015-0888-4

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  • DOI: https://doi.org/10.1007/s00205-015-0888-4

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