Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T11:18:01.881Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal regularity for the pseudo infinity Laplacian

Published online by Cambridge University Press:  12 May 2007

Julio D. Rossi  and
Mariel Saez
Julio D. Rossi
Affiliation:
Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 123, Madrid, Spain, on leave from Departamento de Matemática, FCEyN UBA (1428) Buenos Aires, Argentina; jrossi@dm.uba.ar
Mariel Saez
Affiliation:
Max Planck Institute for Gravitational Physics Albert Einstein Institute Am Mühlenberg 1, 14476 Golm, Germany; mariel.saez@aei.mpg.de
Get access

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.

Keywords

Viscosity solutionsoptimal regularitypseudo infinity Laplacian
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 2 , April 2007 , pp. 294 - 304
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aronsson, G., Extensions of functions satisfiying Lipschitz conditions. Ark. Math. 6 (1967) 551–561. CrossRef
Aronsson, G., Crandall, M.G. and Juutinen, P., A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. 41 (2004) 439–505. CrossRef
Barles, G. and Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Comm. Part. Diff. Eq. 26 (2001) 2323–2337. CrossRef
Belloni, M. and Kawohl, B., The pseudo-p-Laplace eigenvalue problem and viscosity solutions as $p\to \infty$ . ESAIM: COCV 10 (2004) 28–52. CrossRef
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).
Bouchitte, G., Buttazzo, G. and De Pasquale, L., A $p-$ laplacian approximation for some mass optimization problems. J. Optim. Theory Appl. 118 (2003) 125. CrossRef
Crandall, M.G., Ishii, H. and Lions, P.L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. CrossRef
Crandall, M.G., Evans, L.C. and Gariepy, R.F., Optimal Lipschitz extensions and the infinity Laplacian. Calc. Var. PDE 13 (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.
Jensen, R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Rational Mech. Anal. 123 (1993) 51–74. CrossRef
Savin, O., C 1 regularity for infinity harmonic functions in two dimensions. Arch. Rational Mech. Anal. 176 (2005) 351–361. CrossRef