$C^1$-regularity of the Aronsson equation in $R^2$

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1. [1] Aronsson G., Minimization problems for the functional sup $xF(x,f\left(x\right),f(x\left)\right)$, Ark. Mat.6 (1965) 33-53. Zbl0156.12502MR196551
2. [2] Aronsson G., Minimization problems for the functional sup $xF(x,f\left(x\right),f(x\left)\right)$. II, Ark. Mat.6 (1966) 409-431. Zbl0156.12502MR203541
3. [3] Aronsson G., Minimization problems for the functional sup $xF(x,f\left(x\right),f(x\left)\right)$. III, Ark. Mat.7 (1969) 509-512. Zbl0181.11902MR240690
4. [4] Aronsson G., Extension of functions satisfying Lipschitz conditions, Ark. Mat.6 (1967) 551-561. Zbl0158.05001MR217665
5. [5] Aronsson G., On the partial differential equation $u_x^2{u}_{xx}+2u_x{u}_y{u}_{xy}+u_y^2{u}_{yy}=0$, Ark. Mat.7 (1968) 395-425. Zbl0162.42201MR237962
6. [6] Aronsson G., On certain singular solutions of the partial differential equation $u_x^2{u}_{xx}+2u_x{u}_y{u}_{xy}+u_y^2{u}_{yy}=0$, Manuscripta Math.47 (1–3) (1984) 133-151. Zbl0551.35018MR744316
7. [7] Aronsson G., Crandall M., Juutinen P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.)41 (4) (2004) 439-505, (electronic). Zbl1150.35047MR2083637
8. [8] Barron N., Viscosity solutions and analysis in $L^{\infty }$, in: Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, NATO Sci. Ser. C Math. Phys. Sci., vol. 528, Kluwer Acad. Publ., Dordrecht, 1999, pp. 1-60. Zbl0973.49024MR1695005
9. [9] Barron N., Jensen R., Minimizing the $L^{\infty }$-norm of the gradient with an energy constraint, Comm. Partial Differential Equations30 (10–12) (2005) 1741-1772. Zbl1105.35028MR2182310
10. [10] Barron N., Jensen R., Wang C.Y., Lower semicontinuity of $L^{\infty }$ functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire18 (4) (2001) 495-517. Zbl1034.49008MR1841130
11. [11] Barron N., Jensen R., Wang C.Y., The Euler equation and absolute minimizers of $L^{\infty }$ functionals, Arch. Ration. Mech. Anal.157 (4) (2001) 255-283. Zbl0979.49003MR1831173
12. [12] Crandall M., An efficient derivation of the Aronsson equation, Arch. Ration. Mech. Anal.167 (4) (2003) 271-279. Zbl1090.35067MR1981858
13. [13] M. Crandall, The infinity-Laplace equation and the elements of calculus of variations in L-infinity, in: CIME Lecture Notes, in press. 
14. [14] M. Crandall, The continuous gradient flow for the infinity-Laplace equation, Private communication. 
15. [15] M. Crandall, L.C. Evans, A remark on infinity harmonic functions, in: Proceedings of the USA–Chile Workshop on Nonlinear Analysis, Via del Mar-Valparaiso, 2000 (electronic), Electron. J. Differ. Equ. Conf., 6, pp. 123–129. Zbl0964.35061MR1804769
16. [16] Crandall M., Evans C., Gariepy R., Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations13 (2) (2001) 123-139. Zbl0996.49019MR1861094
17. [17] Crandall M., Ishii H., Lions P.L., User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.)27 (1) (1992) 1-67. Zbl0755.35015MR1118699
18. [18] Gariepy R., Wang C.Y., Yu Y., Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers, Comm. Partial Differential Equations31 (7–9) (2006) 1027-1046. Zbl1237.35052MR2254602
19. [19] Jensen R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal.123 (1) (1993) 51-74. Zbl0789.35008MR1218686
20. [20] Morrey C.B., Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math.2 (1952) 25-53. Zbl0046.10803MR54865
21. [21] Savin O., $C^1$-regularity for infinity harmonic functions in dimensions two, Arch. Ration. Mech. Anal.176 (3) (2005) 351-361. Zbl1112.35070MR2185662
22. [22] Yu Y., $L^{\infty }$-variational problems and the Aronsson equations, Arch. Ration. Mech. Anal.182 (1) (2006) 153-180. Zbl1130.49025MR2247955

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