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  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 & 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.   Google Scholar

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

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

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

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

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

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[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.  Google Scholar

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.   Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[31]

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

[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.  Google Scholar

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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