A note on the Harnack inequality for elliptic equations in divergence form
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- by Dongsheng Li and Kai Zhang
- Proc. Amer. Math. Soc. 145 (2017), 135-137
- DOI: https://doi.org/10.1090/proc/13174
- Published electronically: June 10, 2016
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Abstract:
In 1957, De Giorgi proved the Hölder continuity for elliptic equations in divergence form and Moser gave a new proof in 1960. In the next year, Moser obtained the Harnack inequality. In this note, we point out that the Harnack inequality was hidden in De Giorgi’s work.References
- Ennio De Giorgi, Selected papers, Springer-Verlag, Berlin, 2006. Edited by Luigi Ambrosio, Gianni Dal Maso, Marco Forti, Mario Miranda and Sergio Spagnolo. MR 2229237, DOI 10.1007/978-3-642-41496-1
- Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
- Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
- E. DiBenedetto, Harnack estimates in certain function classes, Atti Sem. Mat. Fis. Univ. Modena 37 (1989), no. 1, 173–182. MR 994063
- E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s inequality for degenerate and singular parabolic equations, Springer Monographs in Mathematics, Springer 2011.
- Ugo Gianazza and Vincenzo Vespri, Parabolic De Giorgi classes of order $p$ and the Harnack inequality, Calc. Var. Partial Differential Equations 26 (2006), no. 3, 379–399. MR 2232212, DOI 10.1007/s00526-006-0022-4
- Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 170091, DOI 10.1002/cpa.3160130308
- Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 159138, DOI 10.1002/cpa.3160140329
Bibliographic Information
- Dongsheng Li
- Affiliation: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
- MR Author ID: 647543
- Email: lidsh@mail.xjtu.edu.cn
- Kai Zhang
- Affiliation: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
- MR Author ID: 1098004
- Email: zkzkzk@stu.xjtu.edu.cn
- Received by editor(s): December 14, 2015
- Received by editor(s) in revised form: February 28, 2016
- Published electronically: June 10, 2016
- Additional Notes: This research was supported by NSFC 11171266.
- Communicated by: Joachim Krieger
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 135-137
- MSC (2010): Primary 35J15, 35B65; Secondary 35D30
- DOI: https://doi.org/10.1090/proc/13174
- MathSciNet review: 3565366