Hostname: page-component-6bf8c574d5-j5c6p Total loading time: 0 Render date: 2025-03-06T03:30:56.159Z Has data issue: false hasContentIssue false
  • English
  • Français

REGULARITY OF AML FUNCTIONS IN TWO-DIMENSIONAL NORMED SPACES

Part of: Real functions Partial differential equations

Published online by Cambridge University Press:  20 May 2022

SEBASTIÁN TAPIA-GARCÍA Open the ORCID record for SEBASTIÁN TAPIA-GARCÍA [Opens in a new window]
SEBASTIÁN TAPIA-GARCÍA*
Affiliation:
Institute de Mathématique de Bordeaux, IMB (CNRS UMR 5251), Université de Bordeaux, Cours de la Liberation 351, Talence, France and Departamento de Ingeniería Matemática, CMM (CNRS IRL 2807), Universidad de Chile, Beauchef 851, Santiago, Chile
*
e-mail: stapia@dim.uchile.cl
Get access Rights & Permissions [Opens in a new window]

Abstract

Savin [‘ $\mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations 274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.

Keywords

regularity of Lipschitz functionsabsolutely minimizing Lipschitz functionsfinite normed spaces

MSC classification

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 26A16: Lipschitz (Hölder) classes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 114 , Issue 3 , June 2023 , pp. 406 - 430
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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.)

Footnotes

Communicated by Florica Cirstea

The author was supported by ANID-PFCHA/Doctorado Nacional/2018-21181905 and by CMM (IRL CNRS 2807), Basal grant: AFB170001.

References

Aronsson, G., ‘Extension of functions satisfying Lipschitz conditions’, Ark. Mat. 6 (1967), 551–561.CrossRefGoogle Scholar
Aronsson, G., Crandall, M. and Juutinen, P., ‘A tour of the theory of absolutely minimizing functions’, Bull. Amer. Math. Soc. (N.S.) 41(4) (2004), 439–505.CrossRefGoogle Scholar
Champion, T. and De Pascale, L., ‘Principles of comparison with distance functions for absolute minimizers’, J. Convex Anal. 14(3) (2007), 515–541.Google Scholar
Crandall, M. and Evans, L., ‘A remark on infinity harmonic functions. Proceedings of the USA–Chile Workshop on Nonlinear Analysis (Viña del Mar–Valparaiso, 2000)’, Electron. J. Differ. Equ. Conf. 6 (2001), 123–129.Google Scholar
Crandall, M., Evans, L. and Gariepy, R., ‘Optimal Lipschitz extensions and the infinity Laplacian’, Calc. Var. Partial Differential Equations 13(2) (2001), 123–139.CrossRefGoogle Scholar
Evans, L. and Savin, O., ‘ ${\mathbf{\mathcal{C}}}^{1,\alpha }$ regularity for infinity harmonic functions in two dimensions’, Calc. Var. Partial Differential Equations 32 (2008), 325–347.CrossRefGoogle Scholar
Evans, L. and Smart, C., ‘Adjoint methods for the infinity Laplacian partial differential equation’, Arch. Ration. Mech. Anal. 201 (2011), 87–113.CrossRefGoogle Scholar
Evans, L. and Smart, C., ‘Everywhere differentiability of infinity harmonic functions’, Calc. Var. Partial Differential Equations 42 (2011), 289–299.CrossRefGoogle Scholar
Jensen, R., ‘Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient’, Arch. Ration. Mech. Anal. 123 (1993), 51–74.CrossRefGoogle Scholar
Peng, F., Wang, C. and Zhou, Y., ‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations 274 (2021), 1115–1164.CrossRefGoogle Scholar
Peres, Y., Schramm, O., Sheffield, S. and Wilson, D., ‘Tug-of-war and the infinity Laplacian’, J. Amer. Math. Soc. 22(1) (2009), 167–210.CrossRefGoogle Scholar
Savin, O., ‘ ${C}^1$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3(176) (2005), 351–361.CrossRefGoogle Scholar
Zhang, Y. and Zhou, Y., ‘ ${\mathbf{\mathcal{C}}}^{1,\alpha }$ regularity for infinity harmonic functions in two dimensions’, Proc. Amer. Math. Soc. 148 (2020), 1187–1193.CrossRefGoogle Scholar