Evans, Lawrence C.; Savin, Ovidiu \(C^{1,\alpha}\) regularity for infinity harmonic functions in two dimensions. (English) Zbl 1151.35096 Calc. Var. Partial Differ. Equ. 32, No. 3, 325-347 (2008). The authors of the very interesting paper under review propose a new method for proving \(C^{1,\alpha}\)-regularity of viscosity solutions \(u\) of the infinity Laplacian equation \[ -\Delta_\infty:=u_{x_i}u_{x_j}u_{x_ix_j}=0 \] within an open region \(U\subseteq\mathbb R^n.\) Full details of the proof of \(C^{1,\alpha}\)-regularity in two dimensions are given. The proof for dimensions \(n \geq 3\) depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. Reviewer: Dian K. Palagachev (Bari) Cited in 1 ReviewCited in 60 Documents MSC: 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 35B65 Smoothness and regularity of solutions to PDEs 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions Keywords:infinite Laplace equation; regularity; viscosity solutions PDFBibTeX XMLCite \textit{L. C. Evans} and \textit{O. Savin}, Calc. Var. Partial Differ. Equ. 32, No. 3, 325--347 (2008; Zbl 1151.35096) Full Text: DOI References: [1] Aronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. 41, 439–505 (2004) · Zbl 1150.35047 · doi:10.1090/S0273-0979-04-01035-3 [2] Crandall, M.G., Evans, L.C.: A remark on infinity harmonic functions. In: Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Valparaiso, 2000), vol. 6, pp. 123–129 (electronic). Electronic J. Diff. Equations, Conf. (2001) · Zbl 0964.35061 [3] Crandall, M.G., Evans, L.C., Gariepy, R.F.: Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. Partial Differ. Equ. 13, 123–139 (2001) · Zbl 0996.49019 [4] Evans, L.C., Yu, Y.: Various properties of solutions of the infinity–Laplacian equation. Commun. Partial Differ. Equ. 30, 1401–1428 (2005) · Zbl 1123.35018 · doi:10.1080/03605300500258956 [5] Savin, O.: C 1 regularity for infinity harmonic functions in two dimensions. Arch. Ration. Mech. Anal. 176, 351–361 (2005) · Zbl 1112.35070 · doi:10.1007/s00205-005-0355-8 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.