American Institute of Mathematical Sciences

Advanced
December  2015, 35(12): 6165-6179. doi: 10.3934/dcds.2015.35.6165

The Hessian Sobolev inequality and its extensions

Igor E. Verbitsky 1,

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States

Received  February 2014 Published  May 2015

The Hessian Sobolev inequality of X.-J. Wang, and the Hessian Poincaré inequalities of Trudinger and Wang are fundamental to differential and conformal geometry, and geometric PDE. These remarkable inequalities were originally established via gradient flow methods. In this paper, direct elliptic proofs are given, and extensions to trace inequalities with general measures in place of Lebesgue measure are obtained. The new techniques rely on global estimates of solutions to Hessian equations in terms of Wolff's potentials, and duality arguments making use of a non-commutative inner product on the cone of $k$-convex functions.
Keywords: Wolff potential., k-Hessian, k-convex functions, Hessian Sobolev inequality.
Mathematics Subject Classification: Primary: 35J60, 31C45; Secondary: 46E3.
Citation: Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165
References:
[1]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.

[2]

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

[3]

C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc., 9 (1983), 129-206. doi: 10.1090/S0273-0979-1983-15154-6.

[4]

F. Ferrari, B. Franchi and I. Verbitsky, Hessian inequalities and the fractional Laplacian, J. reine angew. Math., 667 (2012), 133-148. doi: 10.1515/CRELLE.2011.116.

[5]

F. Ferrari and I. Verbitsky, Radial fractional Laplace operators and Hessian inequalities, J. Diff. Eqs., 253 (2012), 244-272. doi: 10.1016/j.jde.2012.03.024.

[6]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. doi: 10.5802/aif.944.

[7]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.

[8]

D. A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9.

[9]

V. G. Maz'ya, Sobolev Spaces, with Applications to Elliptic Partial Differential Equations, 2nd augmented ed., Springer, Berlin, 2011. doi: 10.1007/978-3-642-15564-2.

[10]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.

[11]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal., 256 (2009), 1875-1906. doi: 10.1016/j.jfa.2009.01.012.

[12]

W. Sheng, N. S. Trudinger and X. J. Wang, The Yamabe problem for higher order curvatures, J. Diff. Geom., 77 (2007), 515-553.

[13]

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

[14]

N. S. Trudinger, On new isoperimetric inequalities and symmetrization, J. reine angew. Math., 488 (1997), 203-220. doi: 10.1515/crll.1997.488.203.

[15]

N. S. Trudinger, Weak solutions of Hessian equations, Comm. PDE, 22 (1997), 1251-1261. doi: 10.1080/03605309708821299.

[16]

N. S. Trudinger and X. J. Wang, A Poincaré type inequality for Hessian integrals, Calc. Var. PDE, 6 (1998), 315-328. doi: 10.1007/s005260050093.

[17]

N. S. Trudinger and X. J. Wang, Hessian measures I, Topol. Meth. Nonlin. Anal., 10 (1997), 225-239.

[18]

N. S. Trudinger and X. J. Wang, Hessian measures II, Ann. Math., 150 (1999), 579-604. doi: 10.2307/121089.

[19]

N. S. Trudinger and X. J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369-410. doi: 10.1353/ajm.2002.0012.

[20]

K. Tso, On symmetrization and Hessian equations, J. Anal. Math., 52 (1989), 94-106. doi: 10.1007/BF02820473.

[21]

K. Tso, On a real Monge-Ampère functional, Invent. Math., 101 (1990), 425-448. doi: 10.1007/BF01231510.

[22]

I. E. Verbitsky, Nonlinear potentials and trace inequalities, in The Maz'ya Anniversary Collection, Vol. 2, Operator Theory Adv. Appl., 110, Birkhäuser, 1999, 323-343. doi: 10.1007/978-3-0348-8672-7_18.

[23]

I. E. Verbitsky, Hessian Sobolev and Poincaré inequalities, Oberwolfach Reports, 36 (2011), 2077-2079. doi: 10.4171/OWR/2011/36.

[24]

X. J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., 43 (1994), 25-54. doi: 10.1512/iumj.1994.43.43002.

show all references

References:
[1]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.

[2]

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

[3]

C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc., 9 (1983), 129-206. doi: 10.1090/S0273-0979-1983-15154-6.

[4]

F. Ferrari, B. Franchi and I. Verbitsky, Hessian inequalities and the fractional Laplacian, J. reine angew. Math., 667 (2012), 133-148. doi: 10.1515/CRELLE.2011.116.

[5]

F. Ferrari and I. Verbitsky, Radial fractional Laplace operators and Hessian inequalities, J. Diff. Eqs., 253 (2012), 244-272. doi: 10.1016/j.jde.2012.03.024.

[6]

L. I. Hedberg and T. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble), 33 (1983), 161-187. doi: 10.5802/aif.944.

[7]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.

[8]

D. A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations, Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9.

[9]

V. G. Maz'ya, Sobolev Spaces, with Applications to Elliptic Partial Differential Equations, 2nd augmented ed., Springer, Berlin, 2011. doi: 10.1007/978-3-642-15564-2.

[10]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type, Ann. Math., 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.

[11]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities, J. Funct. Anal., 256 (2009), 1875-1906. doi: 10.1016/j.jfa.2009.01.012.

[12]

W. Sheng, N. S. Trudinger and X. J. Wang, The Yamabe problem for higher order curvatures, J. Diff. Geom., 77 (2007), 515-553.

[13]

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

[14]

N. S. Trudinger, On new isoperimetric inequalities and symmetrization, J. reine angew. Math., 488 (1997), 203-220. doi: 10.1515/crll.1997.488.203.

[15]

N. S. Trudinger, Weak solutions of Hessian equations, Comm. PDE, 22 (1997), 1251-1261. doi: 10.1080/03605309708821299.

[16]

N. S. Trudinger and X. J. Wang, A Poincaré type inequality for Hessian integrals, Calc. Var. PDE, 6 (1998), 315-328. doi: 10.1007/s005260050093.

[17]

N. S. Trudinger and X. J. Wang, Hessian measures I, Topol. Meth. Nonlin. Anal., 10 (1997), 225-239.

[18]

N. S. Trudinger and X. J. Wang, Hessian measures II, Ann. Math., 150 (1999), 579-604. doi: 10.2307/121089.

[19]

N. S. Trudinger and X. J. Wang, On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math., 124 (2002), 369-410. doi: 10.1353/ajm.2002.0012.

[20]

K. Tso, On symmetrization and Hessian equations, J. Anal. Math., 52 (1989), 94-106. doi: 10.1007/BF02820473.

[21]

K. Tso, On a real Monge-Ampère functional, Invent. Math., 101 (1990), 425-448. doi: 10.1007/BF01231510.

[22]

I. E. Verbitsky, Nonlinear potentials and trace inequalities, in The Maz'ya Anniversary Collection, Vol. 2, Operator Theory Adv. Appl., 110, Birkhäuser, 1999, 323-343. doi: 10.1007/978-3-0348-8672-7_18.

[23]

I. E. Verbitsky, Hessian Sobolev and Poincaré inequalities, Oberwolfach Reports, 36 (2011), 2077-2079. doi: 10.4171/OWR/2011/36.

[24]

X. J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J., 43 (1994), 25-54. doi: 10.1512/iumj.1994.43.43002.

[1]

Kods Hassine. Existence and uniqueness of radial solutions for Hardy-Hénon equations involving k-Hessian operators. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022084
[2]

Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1511-1532. doi: 10.3934/dcds.2014.34.1511
[3]

Hedy Attouch, Aïcha Balhag, Zaki Chbani, Hassan Riahi. Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling. Evolution Equations and Control Theory, 2022, 11 (2) : 487-514. doi: 10.3934/eect.2021010
[4]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221
[5]

Limei Dai. Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 579-596. doi: 10.3934/dcds.2020023
[6]

Nina Ivochkina, Nadezda Filimonenkova. On the backgrounds of the theory of m-Hessian equations. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1687-1703. doi: 10.3934/cpaa.2013.12.1687
[7]

João Paulo da Silva, Julio López, Ricardo Dahab. Isogeny formulas for Jacobi intersection and twisted hessian curves. Advances in Mathematics of Communications, 2020, 14 (3) : 507-523. doi: 10.3934/amc.2020048
[8]

Chuanqiang Chen, Li Chen, Xinqun Mei, Ni Xiang. The Neumann problem for a class of mixed complex Hessian equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022049
[9]

Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1
[10]

Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems and Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025
[11]

Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589
[12]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
[13]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[14]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
[15]

Bo Wang, Jiguang Bao. Mirror symmetry for a Hessian over-determined problem and its generalization. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2305-2316. doi: 10.3934/cpaa.2014.13.2305
[16]

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
[17]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377
[18]

Feida Jiang, Xi Chen, Juhua Shi. Nonexistence of entire positive solutions for conformal Hessian quotient inequalities. Electronic Research Archive, 2021, 29 (6) : 4075-4086. doi: 10.3934/era.2021072
[19]

Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067
[20]

Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure and Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (136)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy