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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[1]

Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1511-1532. doi: 10.3934/dcds.2014.34.1511
[2]

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 & Control Theory, 2021  doi: 10.3934/eect.2021010
[3]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221
[4]

Limei Dai. Existence and nonexistence of subsolutions for augmented Hessian equations. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 579-596. doi: 10.3934/dcds.2020023
[5]

Nina Ivochkina, Nadezda Filimonenkova. On the backgrounds of the theory of m-Hessian equations. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1687-1703. doi: 10.3934/cpaa.2013.12.1687
[6]

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
[7]

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

Haixia Liu, Jian-Feng Cai, Yang Wang. Subspace clustering by (k,k)-sparse matrix factorization. Inverse Problems & Imaging, 2017, 11 (3) : 539-551. doi: 10.3934/ipi.2017025
[9]

Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589
[10]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
[11]

Xinqun Mei, Jundong Zhou. The interior gradient estimate of prescribed Hessian quotient curvature equation in the hyperbolic space. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1187-1198. doi: 10.3934/cpaa.2021012
[12]

Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701
[13]

Bo Wang, Jiguang Bao. Mirror symmetry for a Hessian over-determined problem and its generalization. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2305-2316. doi: 10.3934/cpaa.2014.13.2305
[14]

Weisong Dong, Tingting Wang, Gejun Bao. A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1769-1780. doi: 10.3934/cpaa.2016013
[15]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377
[16]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[17]

S. S. Dragomir, I. Gomm. Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 271-278. doi: 10.3934/naco.2012.2.271
[18]

Narciso Román-Roy, Ángel M. Rey, Modesto Salgado, Silvia Vilariño. On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories. Journal of Geometric Mechanics, 2011, 3 (1) : 113-137. doi: 10.3934/jgm.2011.3.113
[19]

Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & 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 & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences