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A note on higher regularity boundary Harnack inequality
The Hessian Sobolev inequality and its extensions
1. | Department of Mathematics, University of Missouri, Columbia, MO 65211, United States |
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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