Blanc, P.; Pinasco, J.P.; Rossi, J.D. "Maximal operators for the P-laplacian family" (2017) Pacific Journal of Mathematics. 287(2):257-295
Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor


We prove existence and uniqueness of viscosity solutions for the problem: max-Δp1u(x), -Δp2u(x) = f(x) in a bounded smooth domain Ω⊂ℝN with u=g on ∂Ω. Here -Δpu=(N+ p)-1|Du|2-pdiv (|Du|p-2Du) is the 1-homogeneous p-Laplacian and we assume that 2 ≤ p1; p2 ≤ ∞. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff ) can choose at every step which are the parameters of the game that regulate the probability of playing a usual tug-ofwar game (without noise) or playing at random. Moreover, the operator max-Δp1u(x), -Δp2u(x) provides a natural analogue with respect to p- Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory. © 2017 Mathematical Sciences Publishers.


Documento: Artículo
Título:Maximal operators for the P-laplacian family
Autor:Blanc, P.; Pinasco, J.P.; Rossi, J.D.
Filiación:Departamento de Matemática, Fceyn Universidad de Buenos Aires Ciudad Universitaria, Pabellòn 1, Buenos Aires, 1428, Argentina
Palabras clave:Dirichlet boundary conditions; Dynamic programming principle; P-Laplacian; Tug-of-war games
Página de inicio:257
Página de fin:295
Título revista:Pacific Journal of Mathematics
Título revista abreviado:Pac. J. Math.


  • Antunović, T., Peres, Y., Sheffield, S., Somersille, S., "Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition" (2012) Comm. Partial Differential Equations, 37 (10), pp. 1839-1869. , MR Zbl
  • Armstrong, S.N., Smart, C.K., "An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions" (2010) Calc. Var. Partial Differential Equations, 37 (3-4), pp. 381-384. , MR Zbl
  • Atar, R., Budhiraja, A., "A stochastic differential game for the inhomogeneous1-Laplace equation" (2010) Ann. Probab, 38 (2), pp. 498-531. , MR Zbl
  • Barles, G., Busca, J., "Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term" (2001) Comm. Partial Differential Equations, 26 (11-12), pp. 2323-2337. , MR Zbl
  • Bjorland, C., Caffarelli, L., Figalli, A., "Non-local gradient dependent operators" (2012) Adv. Math, 230 (4-6), pp. 1859-1894. , MR Zbl
  • Bjorland, C., Caffarelli, L., Figalli, A., "Nonlocal tug-of-war and the infinity fractional Laplacian" (2012) Comm. Pure Appl. Math, 65 (3), pp. 337-380. , MR Zbl
  • Busca, J., Esteban, M.J., Quaas, A., "Nonlinear eigenvalues and bifurcation problems for Pucci's operators" (2005) Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2), pp. 187-206. , MR Zbl
  • Caffarelli, L.A., Cabré, X., (1995) Fully nonlinear elliptic equations, , American Mathematical Society Colloquium Publications 43, American Mathematical Society, Providence, RI MR Zbl
  • Crandall, M.G., Ishii, H., Lions, P.-L., "User's guide to viscosity solutions of second order partial differential equations" (1992) Bull. Amer. Math. Soc.(N.S.), 27 (1), pp. 1-67. , MR Zbl
  • Felmer, P.L., Quaas, A., Tang, M., "On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator" (2006) J. Differential Equations, 226 (1), pp. 80-98. , MR Zbl
  • Hartenstine, D., Rudd, M., "Statistical functional equations and p-harmonious functions" (2013) Adv. Nonlinear Stud, 13 (1), pp. 191-207. , MR Zbl
  • Julin, V., Juutinen, P., "A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation" (2012) Comm. Partial Differential Equations, 37 (5), pp. 934-946. , MR Zbl
  • Juutinen, P., Lindqvist, P., Manfredi, J.J., "On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation" (2001) SIAM J. Math. Anal, 33 (3), pp. 699-717. , MR Zbl
  • Kawohl, B., Manfredi, J., Parviainen, M., "Solutions of nonlinear PDEs in the sense of averages" (2012) J. Math. Pures Appl, 97 (2), pp. 173-188. , (9) MR Zbl
  • Koike, S., Kosugi, T., "Remarks on the comparison principle for quasilinear PDE with no zeroth order terms" (2015) Commun. Pure Appl. Anal, 14 (1), pp. 133-142. , MR Zbl
  • Lindqvist, P., Lukkari, T., "A curious equation involving the 1-Laplacian" (2010) Adv. Calc. Var, 3 (4), pp. 409-421. , MR Zbl
  • Liu, Q., Schikorra, A., (2013) "A game-tree approach to discrete infinity Laplacian with running costs", , preprint arXiv
  • Liu, Q., Schikorra, A., "General existence of solutions to dynamic programming equations" (2015) Commun. Pure Appl. Anal, 14 (1), pp. 167-184. , MR Zbl
  • Llorente, J.G., "A note on unique continuation for solutions of the1-mean value property" (2014) Ann. Acad. Sci. Fenn. Math, 39 (1), pp. 473-483. , MR Zbl
  • Llorente, J.G., "Mean value properties and unique continuation" (2015) Commun. Pure Appl. Anal, 14 (1), pp. 185-199. , MR Zbl
  • Lu, G., Wang, P., "Inhomogeneous infinity Laplace equation" (2008) Adv. Math, 217 (4), pp. 1838-1868. , MR Zbl
  • Luiro, H., Parviainen, M., (2015) "Regularity for nonlinear stochastic games", , preprint arXiv
  • Luiro, H., Parviainen, M., Saksman, E., "Harnack's inequality for p-harmonic functions via stochastic games" (2013) Comm. Partial Differential Equations, 38 (11), pp. 1985-2003. , MR Zbl
  • Maitra, A., Sudderth, W., "Borel stochastic games with lim sup payoff" (1993) Ann. Probab, 21 (2), pp. 861-885. , MR Zbl
  • Maitra, A.P., Sudderth, W.D., (1996) Discrete gambling and stochastic games, , Applications of Mathematics 32, Springer, New York MR Zbl
  • Manfredi, J.J., Parviainen, M., Rossi, J.D., "An asymptotic mean value characterization for p-harmonic functions" (2010) Proc. Amer. Math. Soc, 138 (3), pp. 881-889. , MR Zbl
  • Manfredi, J.J., Parviainen, M., Rossi, J.D., "Dynamic programming principle for tug-of-war games with noise" (2012) ESAIM Control Optim. Calc. Var, 18 (1), pp. 81-90. , MR Zbl
  • Manfredi, J.J., Parviainen, M., Rossi, J.D., "On the definition and properties of p-harmonious functions" (2012) Ann. Sc. Norm. Super. Pisa Cl. Sci, 11 (2), pp. 215-241. , (9) MR Zbl
  • Nyström, K., Parviainen, M., "Tug-of-war, market manipulation, and option pricing" Math. Finance, , (online publication December 2014)
  • Peres, Y., Sheffield, S., "Tug-of-war with noise: a game-theoretic view of the p-Laplacian" (2008) Duke Math. J, 145 (1), pp. 91-120. , MR Zbl
  • Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B., "Tug-of-war and the infinity Laplacian" (2009) J. Amer. Math. Soc, 22 (1), pp. 167-210. , MR Zbl
  • Quaas, A., Sirakov, B., "Existence results for nonproper elliptic equations involving the Pucci operator" (2006) Comm. Partial Differential Equations, 31 (7-9), pp. 987-1003. , MR Zbl
  • Ruosteenoja, E., "Local regularity results for value functions of tug-of-war with noise and running payoff" (2016) Adv. Calc. Var, 9 (1), pp. 1-17. , MR Zbl


---------- APA ----------
Blanc, P., Pinasco, J.P. & Rossi, J.D. (2017) . Maximal operators for the P-laplacian family. Pacific Journal of Mathematics, 287(2), 257-295.
---------- CHICAGO ----------
Blanc, P., Pinasco, J.P., Rossi, J.D. "Maximal operators for the P-laplacian family" . Pacific Journal of Mathematics 287, no. 2 (2017) : 257-295.
---------- MLA ----------
Blanc, P., Pinasco, J.P., Rossi, J.D. "Maximal operators for the P-laplacian family" . Pacific Journal of Mathematics, vol. 287, no. 2, 2017, pp. 257-295.
---------- VANCOUVER ----------
Blanc, P., Pinasco, J.P., Rossi, J.D. Maximal operators for the P-laplacian family. Pac. J. Math. 2017;287(2):257-295.