-
Previous Article
A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions
- DCDS Home
- This Issue
-
Next Article
2-manifolds and inverse limits of set-valued functions on intervals
On the Dirichlet problem for fully nonlinear elliptic equations on annuli of metric cones
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China |
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.
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. Caffarelli, L. 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. Caffarelli, L. 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. Caffarelli, J. Kohn, L. 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. Fang, M.-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. Futaki, H. 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. Gauntlett, D. Martelli, J. 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. Godlinski, W. 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. Guan, S.-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. Guan, N. 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. Hoffman, H. 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. Hou, X.-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. Ivochkina, N. 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. Martelli, J. 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. Caffarelli, L. 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. Caffarelli, L. 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. Caffarelli, J. Kohn, L. 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. Fang, M.-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. Futaki, H. 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. Gauntlett, D. Martelli, J. 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. Godlinski, W. 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. Guan, S.-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. Guan, N. 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. Hoffman, H. 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. Hou, X.-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. Ivochkina, N. 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. Martelli, J. 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. |
[1] |
Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2505-2518. doi: 10.3934/cpaa.2020272 |
[2] |
Martino Bardi, Paola Mannucci. On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2006, 5 (4) : 709-731. doi: 10.3934/cpaa.2006.5.709 |
[3] |
Paola Mannucci. The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction. Communications on Pure and Applied Analysis, 2014, 13 (1) : 119-133. doi: 10.3934/cpaa.2014.13.119 |
[4] |
Isabeau Birindelli, Francoise Demengel. The dirichlet problem for singluar fully nonlinear operators. Conference Publications, 2007, 2007 (Special) : 110-121. doi: 10.3934/proc.2007.2007.110 |
[5] |
Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731 |
[6] |
Byung-Soo Lee. Existence and convergence results for best proximity points in cone metric spaces. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 133-140. doi: 10.3934/naco.2014.4.133 |
[7] |
Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213 |
[8] |
Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070 |
[9] |
Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 |
[10] |
Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 |
[11] |
Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial and Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259 |
[12] |
Yafeng Li, Guo Sun, Yiju Wang. A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 529-537. doi: 10.3934/naco.2011.1.529 |
[13] |
Michael Kühn. Power- and Log-concavity of viscosity solutions to some elliptic Dirichlet problems. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2773-2788. doi: 10.3934/cpaa.2018131 |
[14] |
Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539 |
[15] |
Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 |
[16] |
D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499 |
[17] |
Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145 |
[18] |
Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148 |
[19] |
Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107 |
[20] |
Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]