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\(C^{1,\alpha}\) regularity for infinity harmonic functions in two dimensions. (English) Zbl 1151.35096

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.

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
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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
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