Abstract
We initiate the study of fine p-(super)minimizers, associated with p-harmonic functions, on finely open sets in metric spaces, where \(1 < p < \infty \). After having developed their basic theory, we obtain the p-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted Rn. We build this theory in a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
28 June 2022
A Correction to this paper has been published: https://doi.org/10.1007/s11118-022-10027-8
References
Björn, A.: A weak Kellogg property for quasiminimizers. Comment. Math. Helv. 81, 809–825 (2006)
Björn, A.: A regularity classification of boundary points for p-harmonic functions and quasiminimizers. J. Math. Anal. Appl. 338, 39–47 (2008)
Björn, A., Björn, J.: Boundary regularity for p-harmonic functions and solutions of the obstacle problem. J. Math. Soc. Japan 58, 1211–1232 (2006)
Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, vol. 17. European Math. Soc., Zürich (2011)
Björn, A., Björn, J.: Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology. Rev. Mat Iberoam. 31, 161–214 (2015)
Björn, A., Björn, J.: A uniqueness result for functions with zero fine gradient on quasiconnected sets. Ann. Sc. Norm. Super. Pisa Cl. Sci. 21, 293–301 (2020)
Björn, A., Björn, J., Latvala, V.: The weak Cartan property for the p-fine topology on metric spaces. Indiana Univ. Math. J. 64, 915–941 (2015)
Björn, A., Björn, J., Latvala, V.: Sobolev spaces, fine gradients and quasicontinuity on quasiopen sets. Ann. Acad. Sci. Fenn Math. 41, 551–560 (2016)
Björn, A., Björn, J., Latvala, V.: The Cartan, Choquet and Kellogg properties of the fine topology on metric spaces. J. Anal. Math. 135, 59–83 (2018)
Björn, A., Björn, J., Malý, J.: Quasiopen and p-path open sets, and characterizations of quasicontinuity. Potential Anal. 46, 181–199 (2017)
Björn, A., Björn, J., Shanmugalingam, N.: The Dirichlet problem for p-harmonic functions on metric spaces. J. Reine Angew. Math. 556, 173–203 (2003)
Björn, A., Björn, J., Shanmugalingam, N.: The Perron method for p-harmonic functions. J. Differential Equations 195, 398–429 (2003)
Björn, A., Björn, J., Shanmugalingam, N.: Quasicontinuity of Newton–Sobolev functions and density of Lipschitz functions on metric spaces. Houston. J. Math. 34, 1197–1211 (2008)
Björn, J.: Wiener criterion for Cheeger p-harmonic functions on metric spaces. In: Potential Theory in Matsue, Advanced Studies in Pure Mathematics, vol. 44, pp 103–115. Mathematical Society of Japan, Tokyo (2006)
Björn, J.: Fine continuity on metric spaces. Manuscripta Math. 125, 369–381 (2008)
Björn, J., MacManus, P., Shanmugalingam, N.: Fat sets and pointwise boundary estimates for p-harmonic functions in metric spaces. J. Anal. Math. 85, 339–369 (2001)
Bucur, D., Buttazzo, G., Velichkov, B.: Spectral optimization problems with internal constraint. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 477–495 (2013)
Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Ration. Mech. Anal. 122, 183–195 (1993)
Buttazzo, G., Shrivastava, H.: Optimal shapes for general integral functionals. Ann. H. Lebesgue 3, 261–272 (2020)
Cartan, H.: Théorie générale du balayage en potentiel newtonien. Ann. Univ. Grenoble. Sect. Sci. Math. Phys. 22, 221–280 (1946)
Fuglede, B.: The quasi topology associated with a countably subadditive set function. Ann. Inst. Fourier (Grenoble) 21(1), 123–169 (1971)
Fuglede, B.: Finely Harmonic Functions. Springer, Berlin (1972)
Fuglede, B. : Fonctions harmoniques et fonctions finement harmoniques. Ann. Inst. Fourier (Grenoble) 24(4), 77–91 (1974)
Fuglede, B.: Asymptotic paths for subharmonic functions. Math. Ann. 213, 261–274 (1975)
Fuglede, B.: Sur la fonction de Green pour un domaine fin. Ann Inst Fourier (Grenoble) 25(3-4), 201–206 (1975)
Fuglede, B.: Sur les fonctions finement holomorphes. Ann. Inst. Fourier (Grenoble) 31(4), vii, 57–88 (1981)
Fuglede, B.: On the mean value property of finely harmonic and finely hyperharmonic functions. Aequationes Math. 39, 198–203 (1990)
Fusco, N., Mukherjee, S., Zhang, Y.R.-Y.: A variational characterisation of the second eigenvalue of the p-Laplacian on quasi open sets. Proc. Lond. Math. Soc. 119, 579–612 (2019)
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. 2nd edn. Dover, Mineola, NY (2006)
Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.T.: Sobolev Spaces on Metric Measure Spaces. New Mathematical Monographs, vol. 27. Cambridge Univ. Press, Cambridge (2015)
Kilpeläinen, T., Malý, J.: Supersolutions to degenerate elliptic equation on quasi open sets. Comm. Partial Diff. Equ. 17, 371–405 (1992)
Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)
Kinnunen, J., Martio, O.: Nonlinear potential theory on metric spaces. Illinois Math. J. 46, 857–883 (2002)
Kinnunen, J., Shanmugalingam, N.: Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105, 401–423 (2001)
Korte, R.: A Caccioppoli estimate and fine continuity for superminimizers on metric spaces. Ann. Acad. Sci. Fenn. Math. 33, 597–604 (2008)
Lahti, P.: A notion of fine continuity for BV functions on metric spaces. Potential Anal. 46, 279–294 (2017)
Lahti, P.: A new Cartan-type property and strict quasicoverings when p = 1 in metric spaces. Ann. Acad. Sci. Fenn. Math. 43, 1027–1043 (2018)
Lahti, P.: The Choquet and Kellogg properties for the fine topology when p = 1 in metric spaces. J. Math. Pures Appl. 126, 195–213 (2019)
Lahti P.: Superminimizers and a weak Cartan property for p = 1 in metric spaces. J. Anal Math. 140, 55–87 (2020)
Latvala, V.: Finely superharmonic functions of degenerate elliptic equations. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 96 (1994)
Latvala, V.: A theorem on fine connectedness. Potential Anal. 12, 221–232 (2000)
Lindqvist, P., Martio, O.: Two theorems of N. Wiener for solutions of quasilinear elliptic equations. Acta Math. 155, 153–171 (1985)
Lukeš, J., Malý, J.: Fine hyperharmonicity without Axiom D. Math. Ann. 261, 299–306 (1982)
Lukeš, J., Malý, J., Zajíček, L.: Fine Topology Methods in Real Analysis and Potential Theory. Springer, Berlin (1986)
Lyons, T.: Finely holomorphic functions. J. Funct Anal. 37, 1–18 (1980)
Lyons, T.: A theorem in fine potential theory and applications to finely holomorphic functions. J. Funct. Anal. 37, 19–26 (1980)
Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Amer. Math. Soc., Providence, RI (1997)
Maz’ya, V.G.: On the continuity at a boundary point of solutions of quasi-linear elliptic equations. Vestnik Leningrad Univ. Mat. Mekh. Astronom. 25 (13), 42–55 (1970). (Russian), English transl.: Vestnik Leningrad Univ. Math. 3 (1976), 225–242
Mikkonen, P.: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn. Math. Diss. 104 (1996)
Shanmugalingam, N.: Harmonic functions on metric spaces. Illinois J. Math. 45, 1021–1050 (2001)
Wiener, N.: The Dirichlet problem. J. Math. Phys. 3, 127–146 (1924)
Acknowledgements
Part of this research was done during several visits of V. L. to Linköping University.
Funding
Open access funding provided by Linköping University. A. B. and J. B. were supported by the Swedish Research Council, grants 2016-03424 and 2020-04011 resp. 621-2014-3974 and 2018-04106.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Jan Malý (1955–2021).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Björn, A., Björn, J. & Latvala, V. The Dirichlet Problem for p-minimizers on Finely Open Sets in Metric Spaces. Potential Anal 59, 1117–1140 (2023). https://doi.org/10.1007/s11118-022-09996-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-022-09996-7
Keywords
- Dirichlet problem
- Doubling measure
- Fine continuity
- Fine p-minimizer
- Fine p-superminimizer
- Fine supersolution
- Finely open set
- Metric space
- Nonlinear fine potential theory
- Poincaré inequality
- Quasiopen set