Regularity of viscosity solutions of the biased infinity Laplacian equation. (English) Zbl 1524.35251
Summary: In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the \(\beta\)-biased infinity Laplacian equation
\[
\Delta^\beta_\infty u = 0,
\]
where \(\beta \in \mathbb{R}\) is a fixed constant and \(\Delta^\beta_\infty u := \Delta^N_\infty u + \beta |Du|\), which arises from the random game named biased tug-of-war. By studying directly the \(\beta\)-biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35J70 | Degenerate elliptic equations |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
Keywords:
\(\beta\)-biased infinity Laplacian; viscosity solution; exponential cone; Harnack inequality; Lipschitz regularityReferences:
| [1] | T. Antunović Y. Peres, S. Sheffield, and S. Somersille, Tug-of-war and infinity Laplace equa-tion with vanishing Neumann boundary condition, Commun. Partial Differential Equations, 37(10) (2012), 1839-1869. · Zbl 1268.35065 |
| [2] | S. N. Armstrong, C. K. Smart, and S. J. Somersille, An infinity Laplace equation with gradient term and mixed boundary conditions, Proceedings of the American Mathematical Society, 139(5) (2011), 1763-1776. · Zbl 1216.35062 |
| [3] | G. Aronsson, M. G. Crandall, and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc., 41 (2004), 439-505. · Zbl 1150.35047 |
| [4] | G. Barles, and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Commun. Partial Differential Equations, 26(11-12) (2001), 2323-2337. · Zbl 0997.35023 |
| [5] | E. N. Barron, L. C. Evans, and R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc., 360 (2008), 77-101. · Zbl 1125.35019 |
| [6] | T. Bhattacharya, An elementary proof of the Harnack inequality for non-negative infinity-superharmonic functions, Electronic J. Differential Equations, 44 (2001), 1-8. · Zbl 0966.35052 |
| [7] | F. Charro, J. G. Azorero, and J. D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games, Calc. Var. and Partial Differential Equations, 34(3) (2009), 307-320. · Zbl 1173.35459 |
| [8] | M. G. Crandall, H. Ishii, and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27(1) (1992), 1-67. · Zbl 0755.35015 |
| [9] | M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. and Partial Differential Equations, 13 (2001), 123-139. · Zbl 0996.49019 |
| [10] | K. Does, An evolution equation involving the normalized p−Laplacian, Commun. Pur. Appl. Anal., 10(1) (2011), 361-396. · Zbl 1229.35132 |
| [11] | C. Elion, and L. A. Vese, An image decomposition model using the total variation and the infinity Laplacian, Proc. SPIE, 6498(10) (2007), 49-56. |
| [12] | A. Elmoataz, M. Toutain, and D. Tenbrinck, On the p-Laplacian and ∞-Laplacian on graphs with applications in image and data processing, SIAM J. Imag. Sci., 8(4) (2015), 2412-2451. · Zbl 1330.35488 |
| [13] | L. C. Evans, Estimates for smooth absolutely minimizing Lipschitz extensions, Electronic J. Differential Equations, 3 (1993), 1-10. · Zbl 0809.35032 |
| [14] | L. C. Evans, The 1-Laplacian, the ∞-Laplacian and differential games, Perspectives in Non-linear Partial Differential Equations, Contemp. Math., 446, American Mathematical Society, Providence (2007), 245-254. · Zbl 1200.35114 |
| [15] | L. C. Evans, and O. Savin, C 1,α regularity for infinity harmonic functions in two dimensions, Calc. Var. and Partial Differential Equations, 32(3) (2008), 325-347. · Zbl 1151.35096 |
| [16] | L. C. Evans, and C. K. Smart, Adjoint methods for the infinity Laplacian PDE, Arch. Ration. Mech. Anal., 201(1) (2011), 87-113. · Zbl 1257.35089 |
| [17] | L. C. Evans, and C. K. Smart, Everywhere differentiability of infinity harmonic functions, Calc. Var. Partial Differential Equations, 42(1-2) (2011), 289-299. · Zbl 1251.49034 |
| [18] | C. Le Guillot, and C. L. Guyader, Extrapolation of vector fields using the infinity Laplacian and with applications to image segmentation, Commun. Math. Sci., 7(2) (2009), 423-452. · Zbl 1188.35192 |
| [19] | P. Lindqvist, and J. J. Manfredi, The Harnack inequality for ∞-harmonic functions, Electronic J. Differential Equations, 04 (1995), 1-5. · Zbl 0818.35033 |
| [20] | F. Liu, and F. Jiang, Parabolic biased infinity Laplacian equation related to the biased Tug-of-War, Adv. Nonlinear Stud., 19(1) (2019), 89-112. · Zbl 1412.35163 |
| [21] | Q. Liu, and A. Schikorra, General existence of solutions to dynamic programming equations, Commun. Pure Appl. Anal., 14 (2015), 167-184. · Zbl 1326.35395 |
| [22] | R. López-Soriano, J. C. Navarro-Climent, and Julio D. Rossi, The infinity Laplacian with a transport term, J. Math. Anal. Appl., 398 (2013), 752-765. · Zbl 1257.35076 |
| [23] | G. Lu, and P. Wang, A PDE perspective of the normalized infinity Laplacian, Commun. Partial Differential Equations, 33(10) (2008), 1788-1817. · Zbl 1157.35388 |
| [24] | G. Lu, and P. Wang, Inhomogeneous infinity Laplace equation, Adv. Math., 217(4) (2008), 1838-1868. · Zbl 1152.35042 |
| [25] | Y. Peres, G. Pete, and S. Somersille, Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones, Calc. Var. Partial Differential Equations, 38(3-4) (2010), 541-564. · Zbl 1195.91007 |
| [26] | Y. Peres, O. Schramm, S. Sheffield, and D. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc., 22(1) (2009), 167-210. · Zbl 1206.91002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
