149
Views
1
CrossRef citations to date
0
Altmetric
Research Articles

The convergence rate of p-harmonic to infinity-harmonic functions

ORCID Icon
Pages 1323-1339 | Received 10 Mar 2023, Accepted 12 Nov 2023, Published online: 30 Nov 2023

References

  • Aronsson, G. (1967). Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6(6): 551–561. DOI: 10.1007/BF02591928.
  • Aronsson, G., Crandall, M., Juutinen, P. (2004). A tour of the theory of absolutely minimizing functions. Bull. Amer. Math. Soc. 41(4):439–505. DOI: 10.1090/S0273-0979-04-01035-3.
  • Lindqvist, P. (2016). Notes on the Infinity Laplace Equation. Cham: Springer.
  • Bungert, L., Calder, J., Roith, T. (2022). Uniform convergence rates for Lipschitz learning on graphs. IMA J. Numer. Anal. 43(4):2445–2495. DOI: 10.1093/imanum/drac048.
  • Le Gruyer, E. (2007). On absolutely minimizing Lipschitz extensions and PDE Δ∞(u)=0”. Nonlinear Differ. Equ. Appl. 14(1–2):29–55.
  • Roith, T., Bungert, L. (2022). Continuum limit of Lipschitz learning on graphs. Found. Comput. Math. 23:393–431. DOI: 10.1007/s10208-022-09557-9.
  • Juutinen, P., Shanmugalingam, N. (2006). Equivalence of AMLE, strong AMLE, and comparison with cones in metric measure spaces. Math. Nachr. 279(9–10):1083–1098. DOI: 10.1002/mana.200510411.
  • Juutinen, P. (2002). Absolutely minimizing Lipschitz extensions on a metric space. In: Ann. Acad. Sci. Fenn. Math. 27(1):57–68.
  • Juutinen, P., Lindqvist, P., Manfredi, J. J. (1999). The ∞-eigenvalue problem. Arch. Ration. Mech. Anal. 148(2):89–105. DOI: 10.1007/s002050050157.
  • Esposito, L., Kawohl, B., Nitsch, C., Trombetti, C. (2015). The Neumann eigenvalue problem for the ∞-Laplacian. Rendiconti Lincei-Matematica e Applicazioni 26(2):119–134. DOI: 10.4171/RLM/697.
  • Rossi, J. D., Saintier, N. B. C. (2016). On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions. Houston J. Math. 42(2):613–635.
  • Bungert, L., Korolev, Y. (2022). Eigenvalue problems in L∞: optimality conditions, duality, and relations with optimal transport. Commun. Amer. Math. Soc. 2(8):345–373.
  • Sheffield, S., Smart, C. K. (2012). Vector-valued optimal Lipschitz extensions. Commun. Pure Appl. Math. 65(1):128–154. DOI: 10.1002/cpa.20391.
  • Katzourakis, N. I. (2012). L∞ variational problems for maps and the Aronsson PDE system. J. Differ. Equ. 253(7):2123–2139. DOI: 10.1016/j.jde.2012.05.012.
  • Juutinen, P., Parviainen, M., Rossi, J. D. (2016). Discontinuous gradient constraints and the infinity Laplacian. Int. Math. Res. Not. 2016(8):2451–2492. DOI: 10.1093/imrn/rnv214.
  • Bhattacharya, T., DiBenedetto, E., Manfredi, J. (1989). Limits as p→∞ of Δpu=f and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino 47:15–68.
  • Bungert, L. (2023). The inhomogeneous p-Laplacian equation with Neumann boundary conditions in the limit p→∞. Adv. Contin Discrete Models 2023(1):1–17.
  • Lindgren, E., Lindqvist, P. (2021). The gradient flow of infinity-harmonic potentials. Adv. Math. 378:107526. DOI: 10.1016/j.aim.2020.107526.
  • Brustad, K. K., Lindgren, E., Lindqvist, P. (2023). The infinity-Laplacian in smooth convex domains and in a square. arXiv: 2301.09022 [math.AP]. DOI: 10.3934/mine.2023080.
  • Bungert, L., Calder, J., Roith, T. (2022). Ratio convergence rates for Euclidean firstpassage percolation: applications to the graph infinity Laplacian. arXiv: 2210.09023 [math.PR].
  • Smart, C. K. (2010). On the infinity Laplacian and Hrushovski’s fusion. PhD thesis. UC Berkeley.
  • Armstrong, S., Smart, C. (2012). A finite difference approach to the infinity Laplace equation and tug-of-war games. Trans. Amer. Math. Soc. 364(2):595–636. DOI: 10.1090/S0002-9947-2011-05289-X.
  • Adams, R. A., Fournier, J. J. F. (2003). Sobolev Spaces. Amsterdam: Elsevier.
  • Lindqvist, P. (2019). Notes on the Stationary p-Laplace Equation. Cham: Springer.
  • Barles, G., Busca, J. (2001). Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun. Partial Differ. Equ. 26(11–12): 2323–2337. DOI: 10.1081/PDE-100107824.
  • Jensen, R. (1993). Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123:51–74. DOI: 10.1007/BF00386368.
  • Armstrong, S. N., Smart, C. K. (2010). An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions. Calc. Var. Partial Differ. Equ. 37(3):381–384. DOI: 10.1007/s00526-009-0267-9.
  • Lindqvist, P. (1986). On the definition and properties of p-superharmonic functions. J. für die reine und Angew. Math. (Crelles Journal) 1986(365):67–79.
  • Aronsson, G. (1984). On certain singular solutions of the partial differential equation ux2uxx+2uxuyuxy+uy2uyy=0. Manuscr. Math. 47:133–151. DOI: 10.1007/BF01174590.
  • Aronsson, G. (1986). Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞. Manuscr. Math. 56:135–158. DOI: 10.1007/BF01172152.
  • Bjorland, C., Caffarelli, L., Figalli, A. (2012). Nonlocal tug-of-war and the infinity fractional Laplacian. Commun. Pure Appl. Math. 65(3):337–380. DOI: 10.1002/cpa.21379.
  • del Teso, F., Endal, J., Lewicka, M. (2022). On asymptotic expansions for the fractional infinity Laplacian. Asymptotic Anal. 127(3):201–216. DOI: 10.3233/ASY-211686.
  • del Teso, F., Endal, J., Jakobsen, E. R., Luis Vàzquez, J. (2022). Evolution driven by the infinity fractional Laplacian. arXiv: 2210.06414 [math.AP].
  • Chambolle, A., Lindgren, E., Monneau, R. (2012). A Hölder infinity Laplacian. ESAIM: Control Optim. Calc. Var. 18(3):799–835.
  • Holopainen, I., Shanmugalingam, N. (2002). Singular functions on metric measure spaces. Collect. Math. 53:313–332.
  • Björn, A., Björn, J., Lehrbäck, J. (2020). Existence and almost uniqueness for p-harmonic Green functions on bounded domains in metric spaces. J. Differ. Equ. 269(9):6602–6640. DOI: 10.1016/j.jde.2020.04.044.
  • Björn, A., Björn, J., Lehrbäck, J. (2021). Volume growth, capacity estimates, pparabolicity and sharp integrability properties of p-harmonic Green functions. arXiv: 2101.11486 [math.AP].
  • Crandall, M. G. (2008). A visit with the ∞-Laplace equation. Cal. Var. Nonlinear Partial Differ. Equ. 1927:75–122.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.