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The Green’s function for equations with conic metrics

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Abstract

In this paper, we study the existence, uniqueness and properties of the Green’s function for the conic linear elliptic equation. As an application, we give a new proof of the Schauder estimate for conic Monge–Ampère equations.

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References

  1. Brendle, S.: Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. 34, 5727–5766 (2013)

    Article  Google Scholar 

  2. Chen, X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds I–III. J. Am. Math. Soc. 28(1), 183–278 (2015)

    Article  Google Scholar 

  3. Campana, F., Guenancia, H., Păun, M.: Metrics with cone singularities along normal crossing divisiors and holomorphic tensor fields. Ann. Sci. Éc. Norm. Sup. 46, 879–916 (2013)

    Article  Google Scholar 

  4. Chu, J.: \(C^{2,\alpha }\) regularities and estimates for nonlinear elliptic and parabolic equations in geometry. Calc. Var. Par. Differ. Eqs. 55(1) (2016). Art. 8, 20

  5. Chen, X., Wang, Y.: On the long time behaviour of the conical Kähler Ricci flows. J. Reine Angew. Math. 744, 165–199 (2018)

    MathSciNet  MATH  Google Scholar 

  6. Chen, X., Wang, Y.: Bessel functions, heat kernel and the conical Kähler–Ricci flow. J. Funct. Anal. 269(2), 551–632 (2015)

    Article  MathSciNet  Google Scholar 

  7. Chen, X., Wang, Y.: \(C^{2,\alpha }\) estimate for Monge–Ampère equations with Hölder-continuous right hand side. Ann. Glob. Anal. Geom. 49(2), 195–204 (2016)

    Article  Google Scholar 

  8. Chen, X., Wang, Y.: On the regularity problem of complex Monge–Ampère equations with conical singularities. Ann. Inst. Fourier (Grenoble) 67(3), 969–1003 (2017)

    Article  MathSciNet  Google Scholar 

  9. Calamai, S., Zheng, K.: Geodesics in the space of Kähler cone metrics. Am. J. Math. 137(5), 1149–1208 (2015)

    Article  Google Scholar 

  10. Donaldson, S.: Kähler metric with cone singularitues along the divisor, Essays in Math. and its appl., vol. 4979. Springer, Heidelberg (2012)

    Google Scholar 

  11. Evans, L.C.: Partial Differential Equations: Second Edition. American Mathematical Society (2010)

  12. Huang, L.: A \(C^{2,\alpha ,\beta }\) estimate for conic Monge–Ampère equations, Surveys in Geometric Analysis. Seicence Press, Beijing (2017)

    Google Scholar 

  13. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, Classics in Mathematics, Springer, Berlin (2001). Reprint of the 1998 edition

  14. Grüter, M., Widman, K.-O.: The Green function for uniformly elliptic equations. Manuscripta Math. 37(3), 303–342 (1982)

    Article  MathSciNet  Google Scholar 

  15. Guenancia, H., Păun, M.: Conic singularities metrics with priscribed Ricci curvature general cone angles along Normal crossing divisors. J. Differ. Geom. 103, 15–57 (2016)

    Article  Google Scholar 

  16. Guo, B., Song, J.: Schauder estimate for equations with conic metric \(I\). arxiv:1612.00075v1

  17. Jeffres, T., Mazzeo, R., Rubinstein, Y.: Kähler–Einstein metrics with edge singularities. Ann. Math. 183(1), 95–170 (2016)

    Article  MathSciNet  Google Scholar 

  18. Littman, W., Stampacchia, G., Weinberger, H.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (III) 17, 43–77 (1963)

    MathSciNet  MATH  Google Scholar 

  19. Tian, G.: K-stability and Kähler–Einstein metrics. Comm. Pure Appl. Math. 68(7), 1085–1156 (2015)

    Article  MathSciNet  Google Scholar 

  20. Tian, G.: A Third derivative estimate for conic Monge–Ampère equations. Chin. Ann. Math. Ser. B 38(2), 687–694 (2017)

    Article  MathSciNet  Google Scholar 

  21. Wang, X., Wu, Y.: A new proof for the regulaity of Monge–Ampère type equations. J. Differ. Geom. 116, 543–553 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

The first named author would like to thank his advisors X. Ma and X. Zhang for the encouragement and support, and is grateful to Professor X. Zhu for related discussions over the years. He would also like to thank D. Wu and Y. Zhang for many useful discussions. Part of this work was done while the first named author was visiting the Department of Mathematics at the University of British Columbia, supported by the China Scholarship Council (File No. 201906340217). He would like to thank UBC for the hospitality and support.

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Authors and Affiliations

  1. Westlake Institute for Advanced Study (Westlake University), 18 Shilongshan Road, Hangzhou, 310024, Zhejiang Province, People’s Republic of China

    Liding Huang

  2. School of Mathematical Sciences, Peking University, Yiheyuan Road 5, Beijing, 100871, People’s Republic of China

    Bin Zhou

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  1. Liding Huang
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  2. Bin Zhou
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Correspondence to Bin Zhou.

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Communicated by N. Trudinger.

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Partially supported by National Key R&D Program of China SQ2020YFA0712800 and NSFC 11822101.

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Huang, L., Zhou, B. The Green’s function for equations with conic metrics. Calc. Var. 60, 232 (2021). https://doi.org/10.1007/s00526-021-02103-5

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  • DOI: https://doi.org/10.1007/s00526-021-02103-5

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