Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-02-05T08:30:02.318Z Has data issue: false hasContentIssue false
  • English
  • Français

A Hölder infinity Laplacian

Published online by Cambridge University Press:  17 August 2011

Antonin Chambolle ,
Erik Lindgren  and
Régis Monneau
Antonin Chambolle
Affiliation:
CMAP, École Polytechnique, 91128 Palaiseau Cedex, France. [email protected]
Erik Lindgren
Affiliation:
Dept. of Mathematical Sciences, NTNU, 7491 Trondheim, Norway; [email protected]
Régis Monneau
Affiliation:
Université Paris-Est, Cermics, École des Ponts ParisTech, 6-8, avenue Blaise-Pascal, 77455 Marne-la-Vallée Cedex 2, France; [email protected]
Get access

Abstract

In this paper we study the limit as p → ∞ of minimizers of thefractional Ws,p-norms. In particular, weprove that the limit satisfies a non-local and non-linear equation. We also prove theexistence and uniqueness of solutions of the equation. Furthermore, we prove the existenceof solutions in general for the corresponding inhomogeneous equation. By making strong useof the barriers in this construction, we obtain some regularity results.

Keywords

Lipschitz extensionsHölder extensionsinfinity Laplaciannon-local and non-linear equationsviscosity solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 799 - 835
Copyright
© EDP Sciences, SMAI, 2011

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., Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6 (1967) 551–561. Google Scholar
Aronsson, G., On certain singular solutions of the partial differential equation u 2xu xx + 2u xu yu xy + u 2yu yy = 0. Manuscr. Math. 47 (1984) 133–151. Google Scholar
Aronsson, G., Crandall, M.G. and Juutinen, P., A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. (N.S.) 41 (2004) 439–505. Google Scholar
Barles, G., Chasseigne, E. and Imbert, C., On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57 (2008) 213–246. Google Scholar
T. Bhattacharya, E. Di Benedetto and J. Manfredi, Limits as p → ∞ of Δ pu p = f and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino (1989), No. Special Issue (1991) 15–68, Some topics in nonlinear PDEs, Turin (1989).
C. Bjorland, L. Caffarelli and A. Figalli, Non-Local Tug-of-War and the Infinity Fractional Laplacian. Comm. Pure Appl. Math. (2011).
Caffarelli, L.A. and Córdoba, A., An elementary regularity theory of minimal surfaces. Differential Integral Equations 6 (1993) 1–13. Google Scholar
Caselles, V., Morel, J.-M. and Sbert, C., An axiomatic approach to image interpolation. IEEE Trans. Image Process. 7 (1998) 376–386. Google ScholarPubMed
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. (N.S.) 27 (1992) 1–67. Google Scholar
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. preprint arXiv:1104.4345 (2011)
Lebesgue, H., Sur le problème de Dirichlet. Rend. Circ. Mat. Palermo 24 (1907) 371–402. Google Scholar
Lu, G. and Wang, P., Inhomogeneous infinity Laplace equation. Adv. Math. 217 (2008) 1838–1868. Google Scholar
McShane, E.J., Extension of range of functions. Bull. Am. Math. Soc. 40 (1934) 837–842. Google Scholar
F. Memoli, J.-M. Sapiro and P. Thompson, Brain and surface warping via minimizing lipschitz extensions. IMA Preprint Series 2092 (2006).
K. Murota, Discrete convex analysis. SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003).
Oberman, A.M., An explicit solution of the Lipschitz extension problem. Proc. Am. Math. Soc. 136 (2008) 4329–4338. Google Scholar
Whitney, H., Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36 (1934) 63–89. Google Scholar