June  2022, 21(6): 2101-2114. doi: 10.3934/cpaa.2022051

Gagliardo-Nirenberg-Sobolev inequalities on planar graphs

CEREMADE, CNRS, Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75016 Paris, France

Received  October 2021 Published  June 2022 Early access  March 2022

In this paper we study a family of the Gagliardo-Nirenberg-Sobolev interpolation inequalities on planar graphs. We are interested in knowing when the best constants in the inequalities are achieved. The inequalities being equivalent to some minimization problems, we also analyse the set of solutions of the Euler-Lagrange equations satisfied by extremal functions, or equivalently, by minimizers.

Citation: Maria J. Esteban. Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2101-2114. doi: 10.3934/cpaa.2022051
References:
[1]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J., 85 (1996), 253-270.  doi: 10.1215/S0012-7094-96-08511-7.

[2]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de probabilités, XIX, 1983/84, vol. 1123 of Lecture Notes in Math., Springer, Berlin, 1985, 177–206. doi: 10.1007/BFb0075847.

[3]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411-413. 

[4]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 138 (1993), 213-242.  doi: 10.2307/2946638.

[5]

R. D. BenguriaM. Cristina Depassier and M. Loss, Monotonicity of the period of a non linear oscillator, Nonlinear Anal., 140 (2016), 61-68.  doi: 10.1016/j.na.2016.03.004.

[6]

M. F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539.  doi: 10.1007/BF01243922.

[7]

A. CarlottoO. Chodosh and Y. A. Rubinstein, Slowly converging Yamabe flows, Geom. Topol., 19 (2015), 1523-1568.  doi: 10.2140/gt.2015.19.1523.

[8]

L. Chen, A note on Sobolev type inequalities on graphs with polynomial volume growth, Arch. Math. (Basel), 113 (2019), 313-323.  doi: 10.1007/s00013-019-01329-2.

[9]

J. DolbeaultM. J. Esteban and A. Laptev, Spectral estimates on the sphere, Anal. Partial Differ. Equ., 7 (2014), 435-460.  doi: 10.2140/apde.2014.7.435.

[10]

J. DolbeaultM. J. EstebanA. Laptev and M. Loss, One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows, J. Lond. Math. Soc., 90 (2014), 525-550.  doi: 10.1112/jlms/jdu040.

[11]

J. DolbeaultM. J. Esteban and M. Loss, Nonlinear flows and rigidity results on compact manifolds, J. Funct. Anal., 267 (2014), 1338-1363.  doi: 10.1016/j.jfa.2014.05.021.

[12]

E. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères, Bull. Sci. Math., 121 (1997), 71-96. 

[13]

P. Kuchment, Quantum graphs: an introduction and a brief survey, in Analysis on graphs and its applications, vol. 77 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2008, 291–312. doi: 10.1090/pspum/077/2459876.

[14]

J. R. Licois and L. Véron, Un théorème d'annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1337-1342. 

[15]

J. R. Licois and L. Véron, A class of nonlinear conservative elliptic equations in cylinders, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 249-283. 

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. 

[17]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. 

[18]

M. I. Weinstein, On the solitary traveling wave of the generalized Korteweg-de Vries equation, in Nonlinear systems of partial differential equations in applied mathematics, Part 2 (Santa Fe, N.M., 1984), vol. 23 of Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 1986, 23–30.

[19]

K. Yagasaki, Monotonicity of the period function for $u''-u+ u^p=0$ with $p\in {{\mathbb R}}$ and $p\geq 1$, J. Differ. Equ., 255 (2013), 1988-2001.  doi: 10.1016/j.jde.2013.06.002.

show all references

References:
[1]

D. Bakry and M. Ledoux, Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator, Duke Math. J., 85 (1996), 253-270.  doi: 10.1215/S0012-7094-96-08511-7.

[2]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de probabilités, XIX, 1983/84, vol. 1123 of Lecture Notes in Math., Springer, Berlin, 1985, 177–206. doi: 10.1007/BFb0075847.

[3]

D. Bakry and M. Émery, Inégalités de Sobolev pour un semi-groupe symétrique, C. R. Acad. Sci. Paris Sér. I Math., 301 (1985), 411-413. 

[4]

W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math., 138 (1993), 213-242.  doi: 10.2307/2946638.

[5]

R. D. BenguriaM. Cristina Depassier and M. Loss, Monotonicity of the period of a non linear oscillator, Nonlinear Anal., 140 (2016), 61-68.  doi: 10.1016/j.na.2016.03.004.

[6]

M. F. Bidaut-Véron and L. Véron, Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106 (1991), 489-539.  doi: 10.1007/BF01243922.

[7]

A. CarlottoO. Chodosh and Y. A. Rubinstein, Slowly converging Yamabe flows, Geom. Topol., 19 (2015), 1523-1568.  doi: 10.2140/gt.2015.19.1523.

[8]

L. Chen, A note on Sobolev type inequalities on graphs with polynomial volume growth, Arch. Math. (Basel), 113 (2019), 313-323.  doi: 10.1007/s00013-019-01329-2.

[9]

J. DolbeaultM. J. Esteban and A. Laptev, Spectral estimates on the sphere, Anal. Partial Differ. Equ., 7 (2014), 435-460.  doi: 10.2140/apde.2014.7.435.

[10]

J. DolbeaultM. J. EstebanA. Laptev and M. Loss, One-dimensional Gagliardo-Nirenberg-Sobolev inequalities: remarks on duality and flows, J. Lond. Math. Soc., 90 (2014), 525-550.  doi: 10.1112/jlms/jdu040.

[11]

J. DolbeaultM. J. Esteban and M. Loss, Nonlinear flows and rigidity results on compact manifolds, J. Funct. Anal., 267 (2014), 1338-1363.  doi: 10.1016/j.jfa.2014.05.021.

[12]

E. Fontenas, Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères, Bull. Sci. Math., 121 (1997), 71-96. 

[13]

P. Kuchment, Quantum graphs: an introduction and a brief survey, in Analysis on graphs and its applications, vol. 77 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 2008, 291–312. doi: 10.1090/pspum/077/2459876.

[14]

J. R. Licois and L. Véron, Un théorème d'annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes, C. R. Acad. Sci. Paris Sér. I Math., 320 (1995), 1337-1342. 

[15]

J. R. Licois and L. Véron, A class of nonlinear conservative elliptic equations in cylinders, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 249-283. 

[16]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145. 

[17]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. 

[18]

M. I. Weinstein, On the solitary traveling wave of the generalized Korteweg-de Vries equation, in Nonlinear systems of partial differential equations in applied mathematics, Part 2 (Santa Fe, N.M., 1984), vol. 23 of Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 1986, 23–30.

[19]

K. Yagasaki, Monotonicity of the period function for $u''-u+ u^p=0$ with $p\in {{\mathbb R}}$ and $p\geq 1$, J. Differ. Equ., 255 (2013), 1988-2001.  doi: 10.1016/j.jde.2013.06.002.

Figure 1.  Phase plane for the dynamical system $ v'=u,\; -u'+v= |v|^{p-2}v\;\; \mbox{in}\;\; {\mathbb R} $, with $ p=3 $
Figure 2.  Solution of the Kirchhoff-ODE system for $ p=3 $, and edges of lengths $ 1, 5 $ and $ +\infty $. The dotted curves correspond to the functions $ v_1 $, the dot-dashed to $ v_2 $ and the full line to $ v_0 $. The dots in the graphics correspond in the phase plane to the values of $ v_0, v_1 $ and $ v_2 $ at the origin: the abscissas of the three dots coincide and the sum of the outward derivatives, corresponding to $ - $ the sum of the ordinates of the dots, is equal to $ 0 $
Figure 3.  Solution of the Kirchhoff-ODE system for $ p=3 $, and edges of lengths $ 1, 2 $ and $ 5 $. The dotted curve corresponds to $ v_1 $ ($ \ell_1=1 $), the dot-dashed one to $ v_3 $ ($ \ell_3=5 $) and the full line to $ v_2 $ ($ \ell_2=2 $). We observe that this solution is near the solution to the $ 1, 5, +\infty $ problem on the right of Figure 2. On the contrary, it is difficult to imagine a solution near the one on the left of Figure 2 for $ \ell_2=2 $. Indeed, if $ \ell_2 $ were larger, we would be able to find one such solution, making the number of solutions larger than or equal to $ 2 $. We see this in the next figure, where $ \ell_2= 4 $
Figure 4.  Solution of the Kirchhoff-ODE system for $ p=3 $, and edges of lengths $ 1, 4 $ and $ 5 $. The dotted curves correspond to the functions $ v_1 $ ($ \ell_1=1 $), the dot-dashed to $ v_3 $ ($ \ell_3=5 $) and the full line to $ v_2 $ ($ \ell_2=4 $). The dots in the graphics correspond to the origin : the values of $ v_1(0), v_2(0) $ and $ v_3(0) $ (the abscissas of the three dots) coincide and the sum of the outward derivatives, corresponding to $ - $ the sum of the ordinates of the dots, is equal to $ 0 $. Note that the figure on the left has the same structure as the left figure on Figure 2, and the same analogy is found for the figures on the right
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