November  2017, 37(11): 5707-5730. doi: 10.3934/dcds.2017247

On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones

School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received  May 2017 Published  July 2017

Fund Project: Research supported in part by NSF in China, No. 11571288, No. 11671330, No. 11571332, No. 11625106 and No. 11131007.

In this paper, we study a class of fully nonlinear elliptic equations on annuli of metric cones constructed from closed Sasakian manifolds and derive the a priori estimates assuming the existence of subsolutions. Moreover, such a priori estimates can be applied to certain degenerate equations. A condition for the solvability of Dirichlet problem for non-degenerate fully nonlinear elliptic equations is discovered. Furthermore, we also discuss degenerate equations.

Citation: Chunhui Qiu, Rirong Yuan. On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5707-5730. doi: 10.3934/dcds.2017247
References:
[1]

T. Aubin, Équations du type Monge-Ampère sur les variétés Kähleriennes compactes, (French), Bull. Sci. Math., 102 (1978), 63-95. 

[2]

E. Bedford and B. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37 (1976), 1-4.  doi: 10.1007/BF01418826.

[3]

Z. Blocki, On geodesics in the space of Kähler metrics, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 37 (1976), 1-44. 

[4]

C. Boyer and K. Galicki, Sasakian Geometry, Oxford: Oxford Mathematical Monographs, Oxford University press, 2008.

[5]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ: Functions of eigenvalues of the Hessians, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.

[6]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Rev. Mat. Iberoamericana, 2 (1986), 19-27.  doi: 10.4171/RMI/23.

[7]

L. CaffarelliJ. KohnL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅱ. Complex Monge-Ampère, and uniformaly elliptic, equations, Comm. Pure Applied Math., 38 (1985), 209-252.  doi: 10.1002/cpa.3160380206.

[8]

E. Calabi, The Space of Kähler Metrics, Proc. Internat. Congress of Mathematicians, Amsterdam, Holland. 1954.

[9]

X.-X. Chen, The space of Kähler metrics, J. Diff. Geom., 56 (2000), 189-234.  doi: 10.4310/jdg/1090347643.

[10]

X.-X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852.  doi: 10.4310/CAG.2004.v12.n4.a4.

[11]

T. Collins, A. Jacob and S. -T. Yau, $(1, 1)$ forms with special Lagrangian type: A priori estimates and algebraic obstructions, arXiv: 1508.01934.

[12]

S.-K. Donaldson, Symmeric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, American Mathematical Society Translations: Series 2, 196 . Providence, RI: American Mmathematical Society, 45 (1999), 13-33.  doi: 10.1090/trans2/196/02.

[13]

S.-K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 45 (1999), 13-33.  doi: 10.4310/AJM.1999.v3.n1.a1.

[14]

S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, American Journal of Mathematics, 139 (2017), 403-415, arXiv: 1203.3995. doi: 10.1353/ajm.2017.0009.

[15]

L. Evans, Classical solutions of fully nonlinear convex, second order elliptic equations, Comm. Pure Applied Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.

[16]

H. FangM.-J. Lai and X.-N. Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220.  doi: 10.1515/crelle.2011.027.

[17]

J.-X. Fu, On non-Kähler Calabi-Yau threefolds with Balanced metrics, Proc. Internat. Congress of Mathematicians, Hyderabad, India. New Delhi: Hindustan Book Agency, (2010), 705-716. 

[18]

A. FutakiH. Ono and G.-F. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Diff. Geom., 83 (2009), 585-635.  doi: 10.4310/jdg/1264601036.

[19]

P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518.  doi: 10.1007/BF01455968.

[20]

J. GauntlettD. MartelliJ. Sparks and D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys., 8 (2004), 987-1000.  doi: 10.4310/ATMP.2004.v8.n6.a3.

[21]

M. GodlinskiW. Kopczynski and P. Nurowski, Locally Sasakian manifolds, Classical quantum gravity, 17 (2000), 105-115.  doi: 10.1088/0264-9381/17/18/101.

[22]

B. Guan, The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function, Comm. Anal. Geom., 6 (1998), 687-703.  doi: 10.4310/CAG.1998.v6.n4.a3.

[23]

B. Guan, Second order estimates and regularity for fully nonlinear ellitpic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.

[24]

B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds, preprint.

[25]

B. Guan and Q. Li, The Dirichlet problem for a complex Monge-Ampère type equation on Hermitian manifolds, Adv. Math., 246 (2013), 351-367.  doi: 10.1016/j.aim.2013.07.006.

[26]

B. Guan and X. -L. Nie, Fully nonlinear elliptic equations on Hermitian manifolds, preprint.

[27]

B. Guan and W. Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc. Var. PDE., 54 (2013), 901-916.  doi: 10.1007/s00526-014-0810-1.

[28]

B. GuanS.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE., 8 (2015), 1145-1164.  doi: 10.2140/apde.2015.8.1145.

[29]

B. Guan and J. Spruck, Boundary-value problems on $\mathbb{S}^{n}$ for surfaces of constant Gauss curvature, Ann. Math., 138 (1993), 601-624.  doi: 10.2307/2946558.

[30]

P.-F. Guan, The extremal function associated to intrinsic norms, Ann. Math., 156 (2002), 197-211.  doi: 10.2307/3597188.

[31]

P.-F. GuanN. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104.  doi: 10.1007/BF02392824.

[32]

P.-F. Guan and X. Zhang, A geodesic equation in the space of Sasake metrics, Geometry and Analysis, Adv. Lect. Math., Somerville: International Press, 17 (2011), 303-318. 

[33]

P.-F. Guan and X. Zhang, Regularity of the geodesic equation in the space of Sasake metrics, Adv. Math., 230 (2012), 321-371.  doi: 10.1016/j.aim.2011.12.002.

[34]

D. HoffmanH. Rosenberg and J. Spruck, Boundary value problems for surfaces of constant Gauss Curvature, Comm. Pure Applied Math., 45 (1992), 1051-1062.  doi: 10.1002/cpa.3160450807.

[35]

Z. HouX.-N. Ma and D.-M. Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett., 17 (2010), 547-561.  doi: 10.4310/MRL.2010.v17.n3.a12.

[36]

N. Ivochkina, The integral method of barrier functions and the Dirichlet problem for equations with operators of the Monge-Ampère type, (Russian)Mat. Sb. (N.S.), 112 (1980), 193-206; English transl.: Math. USSR Sb., 40 (1981), 179-192.

[37]

N. IvochkinaN. Trudinger and X.-J. Wang, The Dirichlet problem for degenerate Hessian equations, Comm. PDE., 29 (2004), 219-235.  doi: 10.1081/PDE-120028851.

[38]

N. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108. 

[39]

Y.-Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Applied Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.

[40]

Y.-Y. Li, Degenerate conformally invariant fully nonlinear elliptic equations, Arch. Ration. Mech. Anal., 186 (2007), 25-51.  doi: 10.1007/s00205-006-0041-5.

[41]

T. Mabuchi, Some symplectic geometry on Kähler manifolds. I, Osaka J. Math., 24 (1987), 227-252. 

[42]

D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Comm. Math. Phys., 262 (2006), 51-89.  doi: 10.1007/s00220-005-1425-3.

[43]

D. MartelliJ. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Comm. Mathe. Phys., 280 (2008), 611-673.  doi: 10.1007/s00220-008-0479-4.

[44]

D. -H. Phong, S. Picard and X. -W. Zhang, On estimates for the Fu-Yau generalization of a Strominger system, arXiv: 1507.08193. doi: 10.1515/crelle-2016-0052.

[45]

D.-H. Phong and J. Sturm, The Dirichlet problem for degenerate complex Monge-Ampère equations, Comm. Anal. Geom., 18 (2010), 145-170.  doi: 10.4310/CAG.2010.v18.n1.a6.

[46]

D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bulletin de la SMF, 143 (2015), 763-800, arXiv: 1310.3685. doi: 10.24033/bsmf.2704.

[47]

S. Semmes, Complex Monge-Ampère and sympletic manifolds, Amer. J. Math., 114 (1992), 495-550.  doi: 10.2307/2374768.

[48]

J. Song and B. Weinkove, On the convergence and singularities of the J-Flow with applications to the Mabuchi energy, Comm. Pure Applied Math., 61 (2008), 210-229.  doi: 10.1002/cpa.20182.

[49]

W. Sun, Generalized complex Monge-Ampère type equations on closed Hermitian manifolds, arXiv: 1412.8192.

[50]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, arXiv: 1501.02762.

[51]

G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form, arXiv: 1503.04491.

[52]

M. E. Taylor, Partial Differential Equations I, Basic Theory, New York, Berlin, Heidelberg: Applied Mathematical Sciences, 115, Springer-Verlag, 1996. doi: 10.1007/978-1-4684-9320-7.

[53]

G. Tian and S.-T. Yau, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc., 3 (1990), 579-609.  doi: 10.2307/1990928.

[54]

V. Tosatti and B. Weinkove, Hermitian metrics, $(n-1, n-1)$-forms and Monge-Ampère equations, arXiv: 1310.6326.

[55]

N. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.

[56]

S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Applied Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.

show all references

References:
[1]

T. Aubin, Équations du type Monge-Ampère sur les variétés Kähleriennes compactes, (French), Bull. Sci. Math., 102 (1978), 63-95. 

[2]

E. Bedford and B. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math., 37 (1976), 1-4.  doi: 10.1007/BF01418826.

[3]

Z. Blocki, On geodesics in the space of Kähler metrics, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 37 (1976), 1-44. 

[4]

C. Boyer and K. Galicki, Sasakian Geometry, Oxford: Oxford Mathematical Monographs, Oxford University press, 2008.

[5]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ: Functions of eigenvalues of the Hessians, Acta Math., 155 (1985), 261-301.  doi: 10.1007/BF02392544.

[6]

L. CaffarelliL. Nirenberg and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation, Rev. Mat. Iberoamericana, 2 (1986), 19-27.  doi: 10.4171/RMI/23.

[7]

L. CaffarelliJ. KohnL. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. Ⅱ. Complex Monge-Ampère, and uniformaly elliptic, equations, Comm. Pure Applied Math., 38 (1985), 209-252.  doi: 10.1002/cpa.3160380206.

[8]

E. Calabi, The Space of Kähler Metrics, Proc. Internat. Congress of Mathematicians, Amsterdam, Holland. 1954.

[9]

X.-X. Chen, The space of Kähler metrics, J. Diff. Geom., 56 (2000), 189-234.  doi: 10.4310/jdg/1090347643.

[10]

X.-X. Chen, A new parabolic flow in Kähler manifolds, Comm. Anal. Geom., 12 (2004), 837-852.  doi: 10.4310/CAG.2004.v12.n4.a4.

[11]

T. Collins, A. Jacob and S. -T. Yau, $(1, 1)$ forms with special Lagrangian type: A priori estimates and algebraic obstructions, arXiv: 1508.01934.

[12]

S.-K. Donaldson, Symmeric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, American Mathematical Society Translations: Series 2, 196 . Providence, RI: American Mmathematical Society, 45 (1999), 13-33.  doi: 10.1090/trans2/196/02.

[13]

S.-K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math., 45 (1999), 13-33.  doi: 10.4310/AJM.1999.v3.n1.a1.

[14]

S. Dinew and S. Kolodziej, Liouville and Calabi-Yau type theorems for complex Hessian equations, American Journal of Mathematics, 139 (2017), 403-415, arXiv: 1203.3995. doi: 10.1353/ajm.2017.0009.

[15]

L. Evans, Classical solutions of fully nonlinear convex, second order elliptic equations, Comm. Pure Applied Math., 35 (1982), 333-363.  doi: 10.1002/cpa.3160350303.

[16]

H. FangM.-J. Lai and X.-N. Ma, On a class of fully nonlinear flows in Kähler geometry, J. Reine Angew. Math., 653 (2011), 189-220.  doi: 10.1515/crelle.2011.027.

[17]

J.-X. Fu, On non-Kähler Calabi-Yau threefolds with Balanced metrics, Proc. Internat. Congress of Mathematicians, Hyderabad, India. New Delhi: Hindustan Book Agency, (2010), 705-716. 

[18]

A. FutakiH. Ono and G.-F. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Diff. Geom., 83 (2009), 585-635.  doi: 10.4310/jdg/1264601036.

[19]

P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518.  doi: 10.1007/BF01455968.

[20]

J. GauntlettD. MartelliJ. Sparks and D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys., 8 (2004), 987-1000.  doi: 10.4310/ATMP.2004.v8.n6.a3.

[21]

M. GodlinskiW. Kopczynski and P. Nurowski, Locally Sasakian manifolds, Classical quantum gravity, 17 (2000), 105-115.  doi: 10.1088/0264-9381/17/18/101.

[22]

B. Guan, The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function, Comm. Anal. Geom., 6 (1998), 687-703.  doi: 10.4310/CAG.1998.v6.n4.a3.

[23]

B. Guan, Second order estimates and regularity for fully nonlinear ellitpic equations on Riemannian manifolds, Duke Math. J., 163 (2014), 1491-1524.  doi: 10.1215/00127094-2713591.

[24]

B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds, preprint.

[25]

B. Guan and Q. Li, The Dirichlet problem for a complex Monge-Ampère type equation on Hermitian manifolds, Adv. Math., 246 (2013), 351-367.  doi: 10.1016/j.aim.2013.07.006.

[26]

B. Guan and X. -L. Nie, Fully nonlinear elliptic equations on Hermitian manifolds, preprint.

[27]

B. Guan and W. Sun, On a class of fully nonlinear elliptic equations on Hermitian manifolds, Calc. Var. PDE., 54 (2013), 901-916.  doi: 10.1007/s00526-014-0810-1.

[28]

B. GuanS.-J. Shi and Z.-N. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, Anal. PDE., 8 (2015), 1145-1164.  doi: 10.2140/apde.2015.8.1145.

[29]

B. Guan and J. Spruck, Boundary-value problems on $\mathbb{S}^{n}$ for surfaces of constant Gauss curvature, Ann. Math., 138 (1993), 601-624.  doi: 10.2307/2946558.

[30]

P.-F. Guan, The extremal function associated to intrinsic norms, Ann. Math., 156 (2002), 197-211.  doi: 10.2307/3597188.

[31]

P.-F. GuanN. Trudinger and X.-J. Wang, On the Dirichlet problem for degenerate Monge-Ampère equations, Acta Math., 182 (1999), 87-104.  doi: 10.1007/BF02392824.

[32]

P.-F. Guan and X. Zhang, A geodesic equation in the space of Sasake metrics, Geometry and Analysis, Adv. Lect. Math., Somerville: International Press, 17 (2011), 303-318. 

[33]

P.-F. Guan and X. Zhang, Regularity of the geodesic equation in the space of Sasake metrics, Adv. Math., 230 (2012), 321-371.  doi: 10.1016/j.aim.2011.12.002.

[34]

D. HoffmanH. Rosenberg and J. Spruck, Boundary value problems for surfaces of constant Gauss Curvature, Comm. Pure Applied Math., 45 (1992), 1051-1062.  doi: 10.1002/cpa.3160450807.

[35]

Z. HouX.-N. Ma and D.-M. Wu, A second order estimate for complex Hessian equations on a compact Kähler manifold, Math. Res. Lett., 17 (2010), 547-561.  doi: 10.4310/MRL.2010.v17.n3.a12.

[36]

N. Ivochkina, The integral method of barrier functions and the Dirichlet problem for equations with operators of the Monge-Ampère type, (Russian)Mat. Sb. (N.S.), 112 (1980), 193-206; English transl.: Math. USSR Sb., 40 (1981), 179-192.

[37]

N. IvochkinaN. Trudinger and X.-J. Wang, The Dirichlet problem for degenerate Hessian equations, Comm. PDE., 29 (2004), 219-235.  doi: 10.1081/PDE-120028851.

[38]

N. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, (Russian)Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108. 

[39]

Y.-Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Applied Math., 43 (1990), 233-271.  doi: 10.1002/cpa.3160430204.

[40]

Y.-Y. Li, Degenerate conformally invariant fully nonlinear elliptic equations, Arch. Ration. Mech. Anal., 186 (2007), 25-51.  doi: 10.1007/s00205-006-0041-5.

[41]

T. Mabuchi, Some symplectic geometry on Kähler manifolds. I, Osaka J. Math., 24 (1987), 227-252. 

[42]

D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Comm. Math. Phys., 262 (2006), 51-89.  doi: 10.1007/s00220-005-1425-3.

[43]

D. MartelliJ. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Comm. Mathe. Phys., 280 (2008), 611-673.  doi: 10.1007/s00220-008-0479-4.

[44]

D. -H. Phong, S. Picard and X. -W. Zhang, On estimates for the Fu-Yau generalization of a Strominger system, arXiv: 1507.08193. doi: 10.1515/crelle-2016-0052.

[45]

D.-H. Phong and J. Sturm, The Dirichlet problem for degenerate complex Monge-Ampère equations, Comm. Anal. Geom., 18 (2010), 145-170.  doi: 10.4310/CAG.2010.v18.n1.a6.

[46]

D. Popovici, Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds, Bulletin de la SMF, 143 (2015), 763-800, arXiv: 1310.3685. doi: 10.24033/bsmf.2704.

[47]

S. Semmes, Complex Monge-Ampère and sympletic manifolds, Amer. J. Math., 114 (1992), 495-550.  doi: 10.2307/2374768.

[48]

J. Song and B. Weinkove, On the convergence and singularities of the J-Flow with applications to the Mabuchi energy, Comm. Pure Applied Math., 61 (2008), 210-229.  doi: 10.1002/cpa.20182.

[49]

W. Sun, Generalized complex Monge-Ampère type equations on closed Hermitian manifolds, arXiv: 1412.8192.

[50]

G. Székelyhidi, Fully non-linear elliptic equations on compact Hermitian manifolds, arXiv: 1501.02762.

[51]

G. Székelyhidi, V. Tosatti and B. Weinkove, Gauduchon metrics with prescribed volume form, arXiv: 1503.04491.

[52]

M. E. Taylor, Partial Differential Equations I, Basic Theory, New York, Berlin, Heidelberg: Applied Mathematical Sciences, 115, Springer-Verlag, 1996. doi: 10.1007/978-1-4684-9320-7.

[53]

G. Tian and S.-T. Yau, Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc., 3 (1990), 579-609.  doi: 10.2307/1990928.

[54]

V. Tosatti and B. Weinkove, Hermitian metrics, $(n-1, n-1)$-forms and Monge-Ampère equations, arXiv: 1310.6326.

[55]

N. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math., 175 (1995), 151-164.  doi: 10.1007/BF02393303.

[56]

S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Applied Math., 31 (1978), 339-411.  doi: 10.1002/cpa.3160310304.

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