Optimal regularity for the pseudo infinity Laplacian

Julio D. Rossi; Mariel Saez

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

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

Abstract

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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.

How to cite

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Rossi, 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

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  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.  Zbl1150.35047
  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.  Zbl0997.35023
  4. M. Belloni and B. Kawohl, The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p . ESAIM: COCV10 (2004) 28–52.  Zbl1092.35074
  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).  Zbl1245.35078
  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.  Zbl1040.49040
  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.  
  8. M.G. Crandall, L.C. Evans and R.F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. PDE13 (2001) 123–139.  Zbl0996.49019
  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.  Zbl0920.49004
  10. R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal.123 (1993) 51–74.  Zbl0789.35008
  11. O. Savin, C1 regularity for infinity harmonic functions in two dimensions. Arch. Rational Mech. Anal.176 (2005) 351–361.  Zbl1112.35070

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