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February  2021, 20(2): 783-799. doi: 10.3934/cpaa.2020290

The anisotropic fractional isoperimetric problem with respect to unconditional unit balls

Andreas Kreuml

Technische Universität Wien, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstraße 8-10/1046, 1040 Vienna, Austria

Received  May 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

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.

Keywords: Fractional perimeters, non-local functionals, isoperimetric inequalities, Pólya-Szegő principles, anisotropic symmetrization.
Mathematics Subject Classification: Primary:49Q20;Secondary:26D15, 46E35.
Citation: Andreas Kreuml. The anisotropic fractional isoperimetric problem with respect to unconditional unit balls. Communications on Pure and Applied Analysis, 2021, 20 (2) : 783-799. doi: 10.3934/cpaa.2020290
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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