Optimal regularity for the pseudo infinity laplacian
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 294-304.

In this paper we find the optimal regularity for viscosity solutions of the pseudo infinity laplacian. We prove that the solutions are locally Lipschitz and show an example that proves that this result is optimal. We also show existence and uniqueness for the Dirichlet problem.

DOI : 10.1051/cocv:2007018
Classification : 35A05, 35B65, 35J15
Mots-clés : viscosity solutions, optimal regularity, pseudo infinity laplacian 
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     title = {Optimal regularity for the pseudo infinity laplacian},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {294--304},
     publisher = {EDP-Sciences},
     volume = {13},
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     doi = {10.1051/cocv:2007018},
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     zbl = {1129.35087},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2007018/}
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Rossi, Julio D.; Saez, Mariel. Optimal regularity for the pseudo infinity laplacian. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 294-304. doi : 10.1051/cocv:2007018. http://archive.numdam.org/articles/10.1051/cocv:2007018/

[1] G. Aronsson, Extensions of functions satisfiying Lipschitz conditions. Ark. Math. 6 (1967) 551-561. | Zbl

[2] 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.

[3] G. Barles and J. Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Comm. Part. Diff. Eq. 26 (2001) 2323-2337. | Zbl

[4] M. Belloni and B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p→∞. ESAIM: COCV 10 (2004) 28-52. | Numdam | Zbl

[5] M. Belloni, B. Kawohl and P. Juutinen, The p-Laplace eigenvalue problem as p→∞ in a Finsler metric. J. Europ. Math. Soc. (to appear).

[6] G. Bouchitte, G. Buttazzo and L. De Pasquale, A p-laplacian approximation for some mass optimization problems. J. Optim. Theory Appl. 118 (2003) 125. | MR | Zbl

[7] 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 (1992) 1-67. | Zbl

[8] M.G. Crandall, L.C. Evans and R.F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. PDE 13 (2001) 123-139. | Zbl

[9] L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999), No. 653. | MR | Zbl

[10] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 (1993) 51-74. | Zbl

[11] O. Savin, C 1 regularity for infinity harmonic functions in two dimensions. Arch. Rational Mech. Anal. 176 (2005) 351-361. | Zbl

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