November  2020, 19(11): 5285-5303. doi: 10.3934/cpaa.2020238

On the symmetry and monotonicity of Morrey extremals

Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395

* Corresponding author

Received  April 2020 Revised  May 2020 Published  November 2020 Early access  September 2020

Fund Project: This work was partially supported by NSF grant DMS-1554130

We employ Clarkson's inequality to deduce that each extremal of Morrey's inequality is axially symmetric and is antisymmetric with respect to reflection about a plane orthogonal to its axis of symmetry. We also use symmetrization methods to show that each extremal is monotone in the distance from its axis and in the direction of its axis when restricted to spheres centered at the intersection of its axis and its antisymmetry plane.

Citation: Ryan Hynd, Francis Seuffert. On the symmetry and monotonicity of Morrey extremals. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5285-5303. doi: 10.3934/cpaa.2020238
References:
[1]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976), 269-296. 

[2]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598. 

[3]

F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math, 134 (1998), 335-361.  doi: 10.1007/s002220050267.

[4]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.  doi: 10.1016/0022-1236(91)90099-Q.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science & Business Media, 2010.

[6]

F. Brock and A. Solynin, An approach to symmetrization via polarization, Trans. Amer. Math. Soc., 352 (2000), 1759-1796.  doi: 10.1090/S0002-9947-99-02558-1.

[7]

E. Carlen and A. Figalli, Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation, Duke Math. J., 162 (2013), 579-625.  doi: 10.1215/00127094-2019931.

[8]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure. Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[9]

A. Cianchi, Sharp Morrey-Sobolev inequalities and the distance from extremals, Trans. Amer. Math. Soc., 360 (2008), 4335-4347.  doi: 10.1090/S0002-9947-08-04491-7.

[10]

A. CianchiN. FuscoF. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc., 11 (2009), 1105-1139.  doi: 10.4171/JEMS/176.

[11]

J. Clarkson, Uniformly convex spaces, Trans. Am. Math. Soc., 40 (1936), 396-414.  doi: 10.2307/1989630.

[12]

D. Cordero-ErausquinB. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182 (2004), 307-332.  doi: 10.1016/S0001-8708(03)00080-X.

[13]

Jean. DolbeaultM. EstebanM. Loss and G. Tarantello, On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlinear Stud., 9 (2009), 713-726.  doi: 10.1515/ans-2009-0407.

[14]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.

[15]

G. Folland, Real Analysis. Modern Techniques and Their Applications. John Wiley & Sons, Inc., New York, 1999.

[16]

N. FuscoF. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. math., 168 (2008), 941-980.  doi: 10.4007/annals.2008.168.941.

[17]

M. Gazzini and R. Musina, Hardy-Sobolev-Maz'ya inequalities: symmetry and breaking symmetry of extremal functions. Commun. Contemp. Math., 11 (2009), 993-1007. doi: 10.1142/S0219199709003636.

[18]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics Vol 5, New York University, 1999.

[19]

R. Hynd and F. Seuffert, Extremal functions for Morrey's inequality. Math. Ann., 2018. doi: 10.1007/s00208-018-1775-8.

[20]

R. Hynd and F. Seuffert, Asymptotic flatness of Morrey extremals. arXiv: 1905.07060.

[21]

E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[22]

C. S. Lin and Z. Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Am. Math. Soc., 132 (2004), 1685-1691.  doi: 10.1090/S0002-9939-04-07245-4.

[23]

F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities, J. Geom. Anal., 15 (2005), 83-121.  doi: 10.1007/BF02921860.

[24]

C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.  doi: 10.2307/1989904.

[25]

R. Neumayer, A note on the strong-form stability for the Sobolev Inequality. Calc. Var. Partial Differ. Equ., 59 (2020), 8pp. doi: 10.1007/s00526-019-1686-x.

[26]

J. Sarvas, Symmetrization of condensers in $n$-space. Ann. Acad. Sci. Fenn. Ser. A. I., 522 (1972), 44 pp. doi: 10.5186/aasfm.1973.522.

[27]

F. Seuffert, A stability result for a family of sharp Gagliardo-Nirenberg inequalities. arXiv: 1610.06869.

[28]

F. Seuffert, An extension of the Bianchi–Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions, J. Funct. Anal., 273 (2017), 3094-3149.  doi: 10.1016/j.jfa.2017.07.001.

[29]

D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differ. Equ., 18 (2003), 57-75.  doi: 10.1007/s00526-002-0180-y.

[30]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pure Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[31]

G. Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear anal., 5 (1997), 177-230. 

[32]

J. Van Schaftingen, Universal approximation of symmetrizations by polarizations, Proc. Amer. Math. Soc., 134 (2006), 177-186.  doi: 10.1090/S0002-9939-05-08325-5.

show all references

References:
[1]

T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976), 269-296. 

[2]

T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598. 

[3]

F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math, 134 (1998), 335-361.  doi: 10.1007/s002220050267.

[4]

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.  doi: 10.1016/0022-1236(91)90099-Q.

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science & Business Media, 2010.

[6]

F. Brock and A. Solynin, An approach to symmetrization via polarization, Trans. Amer. Math. Soc., 352 (2000), 1759-1796.  doi: 10.1090/S0002-9947-99-02558-1.

[7]

E. Carlen and A. Figalli, Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation, Duke Math. J., 162 (2013), 579-625.  doi: 10.1215/00127094-2019931.

[8]

F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure. Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[9]

A. Cianchi, Sharp Morrey-Sobolev inequalities and the distance from extremals, Trans. Amer. Math. Soc., 360 (2008), 4335-4347.  doi: 10.1090/S0002-9947-08-04491-7.

[10]

A. CianchiN. FuscoF. Maggi and A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc., 11 (2009), 1105-1139.  doi: 10.4171/JEMS/176.

[11]

J. Clarkson, Uniformly convex spaces, Trans. Am. Math. Soc., 40 (1936), 396-414.  doi: 10.2307/1989630.

[12]

D. Cordero-ErausquinB. Nazaret and C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182 (2004), 307-332.  doi: 10.1016/S0001-8708(03)00080-X.

[13]

Jean. DolbeaultM. EstebanM. Loss and G. Tarantello, On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities, Adv. Nonlinear Stud., 9 (2009), 713-726.  doi: 10.1515/ans-2009-0407.

[14]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010. doi: 10.1090/gsm/019.

[15]

G. Folland, Real Analysis. Modern Techniques and Their Applications. John Wiley & Sons, Inc., New York, 1999.

[16]

N. FuscoF. Maggi and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. math., 168 (2008), 941-980.  doi: 10.4007/annals.2008.168.941.

[17]

M. Gazzini and R. Musina, Hardy-Sobolev-Maz'ya inequalities: symmetry and breaking symmetry of extremal functions. Commun. Contemp. Math., 11 (2009), 993-1007. doi: 10.1142/S0219199709003636.

[18]

E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. Courant Lecture Notes in Mathematics Vol 5, New York University, 1999.

[19]

R. Hynd and F. Seuffert, Extremal functions for Morrey's inequality. Math. Ann., 2018. doi: 10.1007/s00208-018-1775-8.

[20]

R. Hynd and F. Seuffert, Asymptotic flatness of Morrey extremals. arXiv: 1905.07060.

[21]

E. Lieb, Sharp Constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[22]

C. S. Lin and Z. Q. Wang, Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities, Proc. Am. Math. Soc., 132 (2004), 1685-1691.  doi: 10.1090/S0002-9939-04-07245-4.

[23]

F. Maggi and C. Villani, Balls have the worst best Sobolev inequalities, J. Geom. Anal., 15 (2005), 83-121.  doi: 10.1007/BF02921860.

[24]

C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.  doi: 10.2307/1989904.

[25]

R. Neumayer, A note on the strong-form stability for the Sobolev Inequality. Calc. Var. Partial Differ. Equ., 59 (2020), 8pp. doi: 10.1007/s00526-019-1686-x.

[26]

J. Sarvas, Symmetrization of condensers in $n$-space. Ann. Acad. Sci. Fenn. Ser. A. I., 522 (1972), 44 pp. doi: 10.5186/aasfm.1973.522.

[27]

F. Seuffert, A stability result for a family of sharp Gagliardo-Nirenberg inequalities. arXiv: 1610.06869.

[28]

F. Seuffert, An extension of the Bianchi–Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions, J. Funct. Anal., 273 (2017), 3094-3149.  doi: 10.1016/j.jfa.2017.07.001.

[29]

D. Smets and M. Willem, Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differ. Equ., 18 (2003), 57-75.  doi: 10.1007/s00526-002-0180-y.

[30]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pure Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[31]

G. Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear anal., 5 (1997), 177-230. 

[32]

J. Van Schaftingen, Universal approximation of symmetrizations by polarizations, Proc. Amer. Math. Soc., 134 (2006), 177-186.  doi: 10.1090/S0002-9939-05-08325-5.

Figure 1.  These diagrams illustrate the monotonicity properties of the Morrey extremal satisfying (1.3) and (1.4) for $ n = 2 $. Theorems 1.2 and 1.3 respectively assert that $ u(x^1)\le u(x^2) $ for $ x^1,x^2\in \mathbb{R}^2 $ which are ordered as on the horizontal lines in the top diagram and as on each circle in the bottom diagram
Figure 2.  The spherical cap $ C_{t,\theta} $ of radius $ t $ and opening angle $ \theta $. The cap contains all points $ x \in \mathbb{R}^2 $ such that $ |x| = t $ and $ x_2 > t \cos \theta $
Figure 3.  The shaded region on top represents a (closed) subset $ A \subset \mathbb{R}^2 $. The shaded region on bottom is $ A^\star\subset \mathbb{R}^2 $, the cap symmetrization of $ A $ in the direction of the positive $ x_2 $ axis
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