# Optimal regularity for the pseudo infinity Laplacian

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 2, page 294-304
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topRossi, Julio D., and Saez, Mariel. "Optimal regularity for the pseudo infinity Laplacian." ESAIM: Control, Optimisation and Calculus of Variations 13.2 (2007): 294-304. <http://eudml.org/doc/249934>.

@article{Rossi2007,

abstract = {
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.
},

author = {Rossi, Julio D., Saez, Mariel},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Viscosity solutions; optimal regularity; pseudo infinity Laplacian; viscosity solutions; existence; uniqueness; Dirichlet problem},

language = {eng},

month = {5},

number = {2},

pages = {294-304},

publisher = {EDP Sciences},

title = {Optimal regularity for the pseudo infinity Laplacian},

url = {http://eudml.org/doc/249934},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Rossi, Julio D.

AU - Saez, Mariel

TI - Optimal regularity for the pseudo infinity Laplacian

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/5//

PB - EDP Sciences

VL - 13

IS - 2

SP - 294

EP - 304

AB -
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.

LA - eng

KW - Viscosity solutions; optimal regularity; pseudo infinity Laplacian; viscosity solutions; existence; uniqueness; Dirichlet problem

UR - http://eudml.org/doc/249934

ER -

## References

top- G. Aronsson, Extensions of functions satisfiying Lipschitz conditions. Ark. Math.6 (1967) 551–561.
- 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.
- 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.
- M. Belloni and B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as $p\to \infty $. ESAIM: COCV10 (2004) 28–52.
- M. Belloni, B. Kawohl and P. Juutinen, The p-Laplace eigenvalue problem as $p\to \infty $ in a Finsler metric. J. Europ. Math. Soc. (to appear).
- G. Bouchitte, G. Buttazzo and L. De Pasquale, A $p-$laplacian approximation for some mass optimization problems. J. Optim. Theory Appl.118 (2003) 125.
- 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.
- M.G. Crandall, L.C. Evans and R.F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. PDE13 (2001) 123–139.
- L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc.137 (1999), No. 653.
- R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal.123 (1993) 51–74.
- O. Savin, C1 regularity for infinity harmonic functions in two dimensions. Arch. Rational Mech. Anal.176 (2005) 351–361.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.