Local Boundedness for Quasi-minimizers to Nonhomogeneous Energy Functional on Metric Measure Spaces

Yan DONG, Dong Yan LI

Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (3) : 489-498.

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Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (3) : 489-498. DOI: 10.12386/A2022sxxb0040

Local Boundedness for Quasi-minimizers to Nonhomogeneous Energy Functional on Metric Measure Spaces

  • Yan DONG1, Dong Yan LI2
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Abstract

We study the quasi-minimizers to nonhomogeneous energy functional on metric measure spaces. Assuming that the metric spaces satisfy doubling condition and Poincar\'{e} inequality, local boundedness for quasi-minimizers is obtained by establishing Caccioppoli inequality and De Giorgi iteration.

Key words

metric measure spaces / quasi-minimizers / boundedness

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Yan DONG, Dong Yan LI. Local Boundedness for Quasi-minimizers to Nonhomogeneous Energy Functional on Metric Measure Spaces. Acta Mathematica Sinica, Chinese Series, 2022, 65(3): 489-498 https://doi.org/10.12386/A2022sxxb0040

References

[1] Björn A., Björn J., Nonlinear Potential Theory on Metric Spaces, European Mathematical Society, Zurich, 2011.
[2] Björn A., Marola N., Moser iteration for (quasi) minimizers on metric spaces, Manuscripta Math., 2006, 121: 339–366.
[3] Chen Y. Z., Wu L. C., Second order Elliptic Equations and Elliptic Systems, Amer. Math. Soc., Providence, RI, 1991.
[4] Franchi B., Serra-Cassano F. Gehring’s lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals, Studia Math., 1996, 120(1): 1–22.
[5] Fusco N., Sbordone C., Higher integrability of the gradient of minimizers of functional with nonstandard growth conditions, Comm. Pure Appl. Math., 1990, 43(5): 673–683.
[6] Giannetti F., Napoli A., Regularity results for a new class of functionals with non-standard growth conditions, J. Differ. Equations, 2013, 254: 1280–1305.
[7] Giusti E., Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
[8] Gong J., Manfredi J., Parviainen M., Nonhomogeneous variational problems and quasi-minimizers on metric spaces, Manuscripta Math., 2012, 137(1–2): 247–271.
[9] Hajłlasz P., Sobolev spaces on an arbitrary metric space, Potential Anal., 1996, 5: 403–415.
[10] Kinnunen J., Shanmugalingam N., Regularity of quasi-minimizers on metric spaces, Manuscripta Math., 2001, 105: 401–423.
[11] Mascolo E., Papi G., Local boundedness of minimizers of integrals of the calculus of variations, Ann. Mat. Pura Appl., 1994, 167: 323–339.
[12] Moscariello G., Nania L., Hölder continuity of minimizers of functionals with nonstandard growth conditions, Ricerche Mat., 1992, 40: 259–273.
[13] Shanmugalingam N., Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam., 2000, 16: 243–278.
[14] Tuominen H., Orlicz–Sobolev spaces on metric measure spaces, Ann. Acad. Sci. Fenn-Math. Diss., 2004, 135: 1–86.
[15] Wang H. J., Niu P. C., Local boundedness for minimizers of convex integral functionals in metric measure spaces, Math. Scand., 2020, 126(2): 259–275.
[16] Wang H. J., Niu P. C., Weighted higher order exponential type inequalities in metric spaces and applications, Georgian Math. J., https://doi.org/10.1515/gmj-2020-2059.
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