Armstrong, Scott N.; Smart, Charles K.; Somersille, Stephanie J. An infinity Laplace equation with gradient term and mixed boundary conditions. (English) Zbl 1216.35062 Proc. Am. Math. Soc. 139, No. 5, 1763-1776 (2011). Summary: We obtain existence, uniqueness, and stability results for the modified 1-homogeneous infinity Laplace equation \(-\Delta_\infty u -\beta|Du|=f\), subject to Dirichlet or mixed Dirichlet-Neumann boundary conditions. Our arguments rely on comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference approximation. Cited in 1 ReviewCited in 14 Documents MSC: 35J70 Degenerate elliptic equations 35J75 Singular elliptic equations 91A15 Stochastic games, stochastic differential games 35J25 Boundary value problems for second-order elliptic equations 35B51 Comparison principles in context of PDEs 35B35 Stability in context of PDEs Keywords:infinity Laplace equation; comparison principle PDFBibTeX XMLCite \textit{S. N. Armstrong} et al., Proc. Am. Math. Soc. 139, No. 5, 1763--1776 (2011; Zbl 1216.35062) Full Text: DOI arXiv References: [1] Scott N. Armstrong and Charles K. Smart, An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differential Equations 37 (2010), no. 3-4, 381 – 384. · Zbl 1187.35104 · doi:10.1007/s00526-009-0267-9 [2] -, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc. (in press). · Zbl 1239.91011 [3] E. N. Barron, L. C. Evans, and R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc. 360 (2008), no. 1, 77 – 101. · Zbl 1125.35019 [4] Fernando Charro, Jesus García Azorero, and Julio D. Rossi, A mixed problem for the infinity Laplacian via tug-of-war games, Calc. Var. Partial Differential Equations 34 (2009), no. 3, 307 – 320. · Zbl 1173.35459 · doi:10.1007/s00526-008-0185-2 [5] M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 123 – 139. · Zbl 0996.49019 [6] E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE \Delta _{\infty }(\?)=0, NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 1-2, 29 – 55. · Zbl 1154.35055 · doi:10.1007/s00030-006-4030-z [7] E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal. 29 (1998), no. 1, 279 – 292. · Zbl 0915.46002 · doi:10.1137/S0036141095294067 [8] G. Lu and P. Wang, Infinity Laplace equation with non-trivial right-hand-side, Electron. J. Differential Equations 77 (2010), 1-12. · Zbl 1194.35194 [9] 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 (2010), no. 3, 541-564. · Zbl 1195.91007 [10] Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Tug-of-war and the infinity Laplacian, J. Amer. Math. Soc. 22 (2009), no. 1, 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.