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

The Hessian Sobolev inequality and its extensions

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.
Citation: Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete & Continuous Dynamical Systems - A, 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, (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. doi: 10.1007/BF02392544.

[3]

C. Fefferman, The uncertainty principle,, Bull. Amer. Math. Soc., 9 (1983), 129. 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. 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. 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. 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. doi: 10.1007/BF02392793.

[8]

D. A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1. 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., (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. 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. 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.

[13]

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

[14]

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

[15]

N. S. Trudinger, Weak solutions of Hessian equations,, Comm. PDE, 22 (1997), 1251. 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. doi: 10.1007/s005260050093.

[17]

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

[18]

N. S. Trudinger and X. J. Wang, Hessian measures II,, Ann. Math., 150 (1999), 579. 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. doi: 10.1353/ajm.2002.0012.

[20]

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

[21]

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

[22]

I. E. Verbitsky, Nonlinear potentials and trace inequalities,, in The Maz'ya Anniversary Collection, 110 (1999), 323. doi: 10.1007/978-3-0348-8672-7_18.

[23]

I. E. Verbitsky, Hessian Sobolev and Poincaré inequalities,, Oberwolfach Reports, 36 (2011), 2077. 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. 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, (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. doi: 10.1007/BF02392544.

[3]

C. Fefferman, The uncertainty principle,, Bull. Amer. Math. Soc., 9 (1983), 129. 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. 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. 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. 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. doi: 10.1007/BF02392793.

[8]

D. A. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1. 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., (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. 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. 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.

[13]

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

[14]

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

[15]

N. S. Trudinger, Weak solutions of Hessian equations,, Comm. PDE, 22 (1997), 1251. 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. doi: 10.1007/s005260050093.

[17]

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

[18]

N. S. Trudinger and X. J. Wang, Hessian measures II,, Ann. Math., 150 (1999), 579. 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. doi: 10.1353/ajm.2002.0012.

[20]

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

[21]

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

[22]

I. E. Verbitsky, Nonlinear potentials and trace inequalities,, in The Maz'ya Anniversary Collection, 110 (1999), 323. doi: 10.1007/978-3-0348-8672-7_18.

[23]

I. E. Verbitsky, Hessian Sobolev and Poincaré inequalities,, Oberwolfach Reports, 36 (2011), 2077. 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. doi: 10.1512/iumj.1994.43.43002.

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