Local boundedness for minimizers of convex integral functionals in metric measure spaces
DOI:
https://doi.org/10.7146/math.scand.a-116244Abstract
In this paper we consider the convex integral functional $ I := \int _\Omega {\Phi (g_u)\,d\mu } $ in the metric measure space $(X,d,\mu )$, where $X$ is a set, $d$ is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of $X$, $u$ belongs to the Orlicz-Sobolev space $N^{1,\Phi }(\Omega )$, Φ is an N-function satisfying the $\Delta _2$-condition, $g_u$ is the minimal Φ-weak upper gradient of $u$. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that $(X,d,\mu )$ satisfies the $(1,1)$-Poincaré inequality. The result of this paper can be applied to the Carnot-Carathéodory space spanned by vector fields satisfying Hörmander's condition.
References
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Tuominen, H., Pointwise behaviour of Orlicz-Sobolev functions, Ann. Mat. Pura Appl. (4) 188 (2009), no. 1, 35–59. https://doi.org/10.1007/s10231-008-0065-6
A\"ıssaoui, N., Another extension of Orlicz-Sobolev spaces to metric spaces, Abstr. Appl. Anal. (2004), no. 1, 1–26. https://doi.org/10.1155/S1085337504309012
Björn, A. and Björn, J., Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, vol. 17, European Mathematical Society (EMS), Zürich, 2011. https://doi.org/10.4171/099
Breit, D. and Verde, A., Quasiconvex variational functionals in Orlicz-Sobolev spaces, Ann. Mat. Pura Appl. (4) 192 (2013), no. 2, 255–271. https://doi.org/10.1007/s10231-011-0222-1
Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. https://doi.org/10.1007/s000390050094
Esposito, L., Leonetti, F., and Mingione, G., Higher integrability for minimizers of integral functionals with $(p,q)$ growth, J. Differential Equations 157 (1999), no. 2, 414–438. https://doi.org/10.1006/jdeq.1998.3614
Franchi, B., Lu, G., and Wheeden, R. L., A relationship between Poincaré-type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices (1996), no. 1, 1–14. https://doi.org/10.1155/S1073792896000013
Hajłasz, P., Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), no. 4, 403–415. https://doi.org/10.1007/BF00275475
Hajłasz, P. and Koskela, P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, 101 pp. https://doi.org/10.1090/memo/0688
Heinonen, J. and Koskela, P., Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. https://doi.org/10.1007/BF02392747
Kinnunen, J. and Shanmugalingam, N., Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), no. 3, 401–423. https://doi.org/10.1007/s002290100193
Krasnosel'ski\u ı, M. A. and Ruticki\u ı, J. B., Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961.
Marcellini, P., Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993), no. 2, 296–333. https://doi.org/10.1006/jdeq.1993.1091
Mascolo, E. and Papi, G., Local boundedness of minimizers of integrals of the calculus of variations, Ann. Mat. Pura Appl. (4) 167 (1994), 323–339. https://doi.org/10.1007/BF01760338
Mocanu, M., A Poincaré inequality for Orlicz-Sobolev functions with zero boundary values on metric spaces, Complex Anal. Oper. Theory 5 (2011), no. 3, 799–810. https://doi.org/10.1007/s11785-010-0068-3
Mocanu, M., Calculus with weak upper gradients based on Banach function spaces, Sci. Stud. Res. Ser. Math. Inform. 22 (2012), no. 1, 41–63.
Niu, P. and Wang, H., Gehring's lemma for Orlicz functions in metric measure spaces and higher integrability for convex integral funcationals, Houston J. Math. 44 (2018), no. 3, 941–974.
Shanmugalingam, N., Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243–279. https://doi.org/10.4171/RMI/275
Tuominen, H., Orlicz-Sobolev spaces on metric measure spaces, Ann. Acad. Sci. Fenn. Math. Diss. (2004), no. 135, 86 pp., Dissertation, University of Jyväskylä, Jyväskylä, 2004.
Tuominen, H., Pointwise behaviour of Orlicz-Sobolev functions, Ann. Mat. Pura Appl. (4) 188 (2009), no. 1, 35–59. https://doi.org/10.1007/s10231-008-0065-6
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Published
2020-05-06
How to Cite
Wang, H., & Niu, P. (2020). Local boundedness for minimizers of convex integral functionals in metric measure spaces. MATHEMATICA SCANDINAVICA, 126(2), 259–275. https://doi.org/10.7146/math.scand.a-116244
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