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The degenerate Monge-Ampère equations with the Neumann condition
The anisotropic fractional isoperimetric problem with respect to unconditional unit balls
Technische Universität Wien, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstraße 8-10/1046, 1040 Vienna, Austria |
The minimizers of the anisotropic fractional isoperimetric inequality with respect to a convex body $ K $ in $ \mathbb{R}^n $ are shown to be equivalent to star bodies whenever $ K $ is strictly convex and unconditional. From this a Pólya-Szegő principle for anisotropic fractional seminorms is derived by using symmetrization with respect to star bodies.
References:
[1] |
F. J. Almgren and E. H. Lieb,
Symmetric decreasing rearrangement is sometimes continuous, J. Am. Math. Soc., 2 (1989), 683-773.
doi: 10.2307/1990893. |
[2] |
A. Alvino, V. Ferone, G. Trombetti and P. L. Lions,
Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 275-293.
doi: 10.1016/S0294-1449(97)80147-3. |
[3] |
M. Amar and G. Bellettini,
A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91-133.
doi: 10.1016/S0294-1449(16)30197-4. |
[4] |
L. Ambrosio, G. De Philippis and L. Martinazzi,
Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.
doi: 10.1007/s00229-010-0399-4. |
[5] |
W. Beckner,
Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. USA, 89 (1992), 4816-4819.
doi: 10.1073/pnas.89.11.4816. |
[6] |
J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001.
![]() |
[7] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^s,p$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[8] |
H. Brezis,
How to recognize constant functions. A connection with Sobolev spaces, Uspekhi Mat. Nauk, 57 (2002), 59-74.
doi: 10.1070/RM2002v057n04ABEH000533. |
[9] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer, Cham, 2016.
doi: 10.1007/978-3-319-28739-3. |
[10] |
L. Caffarelli, J. M. Roquejoffre and O. Savin,
Nonlocal minimal surfaces, Commun. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[11] |
A. Cesaroni and M. Novaga,
The isoperimetric problem for nonlocal perimeters, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 425-440.
doi: 10.3934/dcdss.2018023. |
[12] |
J. Dávila,
On an open question about functions of bounded variation, Calc. Var. Partial Differ. Equ., 15 (2002), 519-527.
doi: 10.1007/s005260100135. |
[13] |
A. Di Castro, M. Novaga, B. Ruffini and E. Valdinoci,
Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differ. Equ., 54 (2015), 2421-2464.
doi: 10.1007/s00526-015-0870-x. |
[14] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[15] |
R. L. Frank and R. Seiringer,
Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.
doi: 10.1016/j.jfa.2008.05.015. |
[16] |
D. R. Fulkerson,
Blocking and anti-blocking pairs of polyhedra, Math. Program., 1 (1971), 168-194.
doi: 10.1007/BF01584085. |
[17] |
N. Fusco,
The classical isoperimetric theorem, Rend. Accad. Sci. Fis. Mat. Napoli, 71 (2004), 63-107.
|
[18] |
R. J. Gardner, Geometric Tomography, 2nd edition, Cambridge University Press, New York, 2006.
doi: 10.1017/CBO9781107341029. |
[19] |
R. Hurri-Syrjänen and A. V. Vähäkangas, Characterizations to the fractional Sobolev inequality, in Complex Analysis and Dynamical Systems VII, American Mathematical Society, Providence, RI, 2017.
doi: 10.1090/conm/699/14087. |
[20] |
A. Kreuml and O. Mordhorst,
Fractional Sobolev norms and BV functions on manifolds, Nonlinear Anal., 187 (2019), 450-466.
doi: 10.1016/j.na.2019.06.014. |
[21] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.
doi: 10.1002/sapm197757293. |
[22] |
E. H. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[23] |
M. Ludwig,
Anisotropic fractional perimeters, J. Differ. Geom., 96 (2014), 77-93.
|
[24] |
M. Ludwig,
Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.
doi: 10.1016/j.aim.2013.10.024. |
[25] |
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge University Press, Cambridge, 2012.
doi: 10.1017/CBO9781139108133.![]() ![]() |
[26] |
G. Pólya and G. Szegő, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, N. J., 1951.
![]() |
[27] |
R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, expanded edition, Cambridge University Press, Cambridge, 2014. |
[28] |
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Springer-Verlag, Berlin, 2003. |
[29] |
J. E. Taylor,
Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588.
doi: 10.1090/S0002-9904-1978-14499-1. |
[30] |
J. Van Schaftingen,
Anisotropic symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 539-565.
doi: 10.1016/j.anihpc.2005.06.001. |
[31] |
A. Visintin,
Nonconvex functionals related to multiphase systems, SIAM J. Math. Anal., 21 (1990), 1281-1304.
doi: 10.1137/0521071. |
[32] |
J. Xiao,
Optimal geometric estimates for fractional Sobolev capacities, C. R. Math. Acad. Sci. Paris, 354 (2016), 149-153.
doi: 10.1016/j.crma.2015.10.014. |
show all references
References:
[1] |
F. J. Almgren and E. H. Lieb,
Symmetric decreasing rearrangement is sometimes continuous, J. Am. Math. Soc., 2 (1989), 683-773.
doi: 10.2307/1990893. |
[2] |
A. Alvino, V. Ferone, G. Trombetti and P. L. Lions,
Convex symmetrization and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 275-293.
doi: 10.1016/S0294-1449(97)80147-3. |
[3] |
M. Amar and G. Bellettini,
A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91-133.
doi: 10.1016/S0294-1449(16)30197-4. |
[4] |
L. Ambrosio, G. De Philippis and L. Martinazzi,
Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.
doi: 10.1007/s00229-010-0399-4. |
[5] |
W. Beckner,
Sobolev inequalities, the Poisson semigroup, and analysis on the sphere $S^n$, Proc. Nat. Acad. Sci. USA, 89 (1992), 4816-4819.
doi: 10.1073/pnas.89.11.4816. |
[6] |
J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001.
![]() |
[7] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^s,p$ when $s\uparrow1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[8] |
H. Brezis,
How to recognize constant functions. A connection with Sobolev spaces, Uspekhi Mat. Nauk, 57 (2002), 59-74.
doi: 10.1070/RM2002v057n04ABEH000533. |
[9] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer, Cham, 2016.
doi: 10.1007/978-3-319-28739-3. |
[10] |
L. Caffarelli, J. M. Roquejoffre and O. Savin,
Nonlocal minimal surfaces, Commun. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[11] |
A. Cesaroni and M. Novaga,
The isoperimetric problem for nonlocal perimeters, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 425-440.
doi: 10.3934/dcdss.2018023. |
[12] |
J. Dávila,
On an open question about functions of bounded variation, Calc. Var. Partial Differ. Equ., 15 (2002), 519-527.
doi: 10.1007/s005260100135. |
[13] |
A. Di Castro, M. Novaga, B. Ruffini and E. Valdinoci,
Nonlocal quantitative isoperimetric inequalities, Calc. Var. Partial Differ. Equ., 54 (2015), 2421-2464.
doi: 10.1007/s00526-015-0870-x. |
[14] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[15] |
R. L. Frank and R. Seiringer,
Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.
doi: 10.1016/j.jfa.2008.05.015. |
[16] |
D. R. Fulkerson,
Blocking and anti-blocking pairs of polyhedra, Math. Program., 1 (1971), 168-194.
doi: 10.1007/BF01584085. |
[17] |
N. Fusco,
The classical isoperimetric theorem, Rend. Accad. Sci. Fis. Mat. Napoli, 71 (2004), 63-107.
|
[18] |
R. J. Gardner, Geometric Tomography, 2nd edition, Cambridge University Press, New York, 2006.
doi: 10.1017/CBO9781107341029. |
[19] |
R. Hurri-Syrjänen and A. V. Vähäkangas, Characterizations to the fractional Sobolev inequality, in Complex Analysis and Dynamical Systems VII, American Mathematical Society, Providence, RI, 2017.
doi: 10.1090/conm/699/14087. |
[20] |
A. Kreuml and O. Mordhorst,
Fractional Sobolev norms and BV functions on manifolds, Nonlinear Anal., 187 (2019), 450-466.
doi: 10.1016/j.na.2019.06.014. |
[21] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.
doi: 10.1002/sapm197757293. |
[22] |
E. H. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[23] |
M. Ludwig,
Anisotropic fractional perimeters, J. Differ. Geom., 96 (2014), 77-93.
|
[24] |
M. Ludwig,
Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.
doi: 10.1016/j.aim.2013.10.024. |
[25] |
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge University Press, Cambridge, 2012.
doi: 10.1017/CBO9781139108133.![]() ![]() |
[26] |
G. Pólya and G. Szegő, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, N. J., 1951.
![]() |
[27] |
R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, expanded edition, Cambridge University Press, Cambridge, 2014. |
[28] |
A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Springer-Verlag, Berlin, 2003. |
[29] |
J. E. Taylor,
Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588.
doi: 10.1090/S0002-9904-1978-14499-1. |
[30] |
J. Van Schaftingen,
Anisotropic symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 539-565.
doi: 10.1016/j.anihpc.2005.06.001. |
[31] |
A. Visintin,
Nonconvex functionals related to multiphase systems, SIAM J. Math. Anal., 21 (1990), 1281-1304.
doi: 10.1137/0521071. |
[32] |
J. Xiao,
Optimal geometric estimates for fractional Sobolev capacities, C. R. Math. Acad. Sci. Paris, 354 (2016), 149-153.
doi: 10.1016/j.crma.2015.10.014. |
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