American Institute of Mathematical Sciences

Advanced
September  2014, 6(3): 335-372. doi: 10.3934/jgm.2014.6.335

Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

Joachim Escher 1, and  Boris Kolev 2,

1. 

Institute for Applied Mathematics, University of Hanover, D-30167 Hanover
2. 

Institut de Mathématiques de Marseille, Aix Marseille Université, CNRS, Centrale Marseille, 13453 Marseille, France

Received  November 2013 Revised  March 2014 Published  September 2014

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.
Keywords: Sobolev metrics of fractional order., Euler equation, diffeomorphism group of the circle.
Mathematics Subject Classification: Primary: 58D05; Secondary: 35Q5.
Citation: Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335
References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[2]

M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Global Anal. Geom., 44 (2013), 5.  doi: 10.1007/s10455-012-9353-x.  Google Scholar

[3]

A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996.  doi: 10.1137/050623036.  Google Scholar

[4]

M. Bruveris, The energy functional on the Virasoro-Bott group with the $L^{2}$-metric has no local minima,, Ann. Global Anal. Geom., 43 (2013), 385.  doi: 10.1007/s10455-012-9350-0.  Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002).  doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[7]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[8]

P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation,, Comm. Pure Appl. Math., 38 (1985), 715.  doi: 10.1002/cpa.3160380605.  Google Scholar

[9]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[10]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

[11]

J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Commun. Pure Appl. Anal., 11 (2012), 1407.  doi: 10.3934/cpaa.2012.11.1407.  Google Scholar

[12]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012).  doi: 10.1142/S0219199712500162.  Google Scholar

[13]

L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable,, Mémoires de l'académie des sciences de Berlin, 14 (1765), 154.   Google Scholar

[14]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[15]

F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space,, PhD thesis, (2009).   Google Scholar

[16]

E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds,, ArXiv e-prints, ().  doi: 10.1007/s12220-014-9523-0.  Google Scholar

[17]

E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve,, ArXiv e-prints, ().   Google Scholar

[18]

L. Guieu and C. Roger, L'algèbre et le Groupe de Virasoro,, Les Publications CRM, (2007).   Google Scholar

[19]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.  doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar

[20]

D. D. Holm, J. E. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[21]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498.  doi: 10.1137/0151075.  Google Scholar

[22]

H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol. 226 of Memoirs of the American Mathematical Society,, 1st edition, (2013).  doi: 10.1090/S0065-9266-2013-00676-4.  Google Scholar

[23]

B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617.  doi: 10.1007/s00208-008-0250-3.  Google Scholar

[24]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116.  doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[25]

B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480.  doi: 10.2991/jnmp.2004.11.4.5.  Google Scholar

[26]

S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999).   Google Scholar

[27]

J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere,, J. Geom. Phys., 57 (2007), 2049.  doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[28]

J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[29]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.   Google Scholar

[30]

J. Milnor, Remarks on infinite-dimensional Lie groups,, in Relativity, (1984), 1007.   Google Scholar

[31]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[32]

G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.  doi: 10.1007/PL00012648.  Google Scholar

[33]

S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1.   Google Scholar

[34]

P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics,, 2nd edition, (1993).  doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[36]

H. Poincaré, Sur une nouvelle forme des équations de la mécanique,, C.R. Acad. Sci., 132 (1901), 369.   Google Scholar

[37]

S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337.  doi: 10.1006/jfan.1998.3335.  Google Scholar

[38]

L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space,, Mem. Amer. Math. Soc., 183 (2006).  doi: 10.1090/memo/0861.  Google Scholar

[39]

F. Tiǧlay and C. Vizman, Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications,, Lett. Math. Phys., 97 (2011), 45.  doi: 10.1007/s11005-011-0464-2.  Google Scholar

[40]

H. Triebel, Theory of Function Spaces,, Birkhäuser Boston, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[41]

M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric,, J. Nonlinear Math. Phys., 17 (2010), 7.  doi: 10.1142/S1402925110000544.  Google Scholar

[42]

Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272.  doi: 10.1137/S0036141003425672.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits,, Ann. Inst. Fourier (Grenoble), 16 (1966), 319.  doi: 10.5802/aif.233.  Google Scholar

[2]

M. Bauer, M. Bruveris, P. Harms and P. W. Michor, Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group,, Ann. Global Anal. Geom., 44 (2013), 5.  doi: 10.1007/s10455-012-9353-x.  Google Scholar

[3]

A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996.  doi: 10.1137/050623036.  Google Scholar

[4]

M. Bruveris, The energy functional on the Virasoro-Bott group with the $L^{2}$-metric has no local minima,, Ann. Global Anal. Geom., 43 (2013), 385.  doi: 10.1007/s10455-012-9350-0.  Google Scholar

[5]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[6]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems,, J. Phys. A, 35 (2002).  doi: 10.1088/0305-4470/35/32/201.  Google Scholar

[7]

A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle,, Comment. Math. Helv., 78 (2003), 787.  doi: 10.1007/s00014-003-0785-6.  Google Scholar

[8]

P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation,, Comm. Pure Appl. Math., 38 (1985), 715.  doi: 10.1002/cpa.3160380605.  Google Scholar

[9]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[10]

J. Escher and B. Kolev, The Degasperis-Procesi equation as a non-metric Euler equation,, Math. Z., 269 (2011), 1137.  doi: 10.1007/s00209-010-0778-2.  Google Scholar

[11]

J. Escher, B. Kolev and M. Wunsch, The geometry of a vorticity model equation,, Commun. Pure Appl. Anal., 11 (2012), 1407.  doi: 10.3934/cpaa.2012.11.1407.  Google Scholar

[12]

J. Escher and M. Wunsch, Restrictions on the geometry of the periodic vorticity equation,, Communications in Contemporary Mathematics, 14 (2012).  doi: 10.1142/S0219199712500162.  Google Scholar

[13]

L. P. Euler, Du mouvement de rotation des corps solides autour d'un axe variable,, Mémoires de l'académie des sciences de Berlin, 14 (1765), 154.   Google Scholar

[14]

A. S. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47.  doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[15]

F. Gay-Balmaz, Infinite Dimensional Geodesic Flows and the Universal Teichmüller Space,, PhD thesis, (2009).   Google Scholar

[16]

E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian geometry on infinite-dimensional manifolds,, ArXiv e-prints, ().  doi: 10.1007/s12220-014-9523-0.  Google Scholar

[17]

E. Grong, I. Markina and A. Vasil'ev, Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmüller space and curve,, ArXiv e-prints, ().   Google Scholar

[18]

L. Guieu and C. Roger, L'algèbre et le Groupe de Virasoro,, Les Publications CRM, (2007).   Google Scholar

[19]

R. S. Hamilton, The inverse function theorem of Nash and Moser,, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 65.  doi: 10.1090/S0273-0979-1982-15004-2.  Google Scholar

[20]

D. D. Holm, J. E. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories,, Adv. Math., 137 (1998), 1.  doi: 10.1006/aima.1998.1721.  Google Scholar

[21]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498.  doi: 10.1137/0151075.  Google Scholar

[22]

H. Inci, T. Kappeler and P. Topalov, On the Regularity of the Composition of Diffeomorphisms, vol. 226 of Memoirs of the American Mathematical Society,, 1st edition, (2013).  doi: 10.1090/S0065-9266-2013-00676-4.  Google Scholar

[23]

B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617.  doi: 10.1007/s00208-008-0250-3.  Google Scholar

[24]

B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits,, Adv. Math., 176 (2003), 116.  doi: 10.1016/S0001-8708(02)00063-4.  Google Scholar

[25]

B. Kolev, Lie groups and mechanics: An introduction,, J. Nonlinear Math. Phys., 11 (2004), 480.  doi: 10.2991/jnmp.2004.11.4.5.  Google Scholar

[26]

S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics,, Springer-Verlag, (1999).   Google Scholar

[27]

J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere,, J. Geom. Phys., 57 (2007), 2049.  doi: 10.1016/j.geomphys.2007.05.003.  Google Scholar

[28]

J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129.  doi: 10.1007/s00220-010-1069-9.  Google Scholar

[29]

P. W. Michor and D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms,, Doc. Math., 10 (2005), 217.   Google Scholar

[30]

J. Milnor, Remarks on infinite-dimensional Lie groups,, in Relativity, (1984), 1007.   Google Scholar

[31]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203.  doi: 10.1016/S0393-0440(97)00010-7.  Google Scholar

[32]

G. Misiołek, Classical solutions of the periodic Camassa-Holm equation,, Geom. Funct. Anal., 12 (2002), 1080.  doi: 10.1007/PL00012648.  Google Scholar

[33]

S. Nag and D. Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle,, Osaka J. Math., 32 (1995), 1.   Google Scholar

[34]

P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics,, 2nd edition, (1993).  doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[36]

H. Poincaré, Sur une nouvelle forme des équations de la mécanique,, C.R. Acad. Sci., 132 (1901), 369.   Google Scholar

[37]

S. Shkoller, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics,, J. Funct. Anal., 160 (1998), 337.  doi: 10.1006/jfan.1998.3335.  Google Scholar

[38]

L. A. Takhtajan and L.-P. Teo, Weil-Petersson metric on the universal Teichmüller space,, Mem. Amer. Math. Soc., 183 (2006).  doi: 10.1090/memo/0861.  Google Scholar

[39]

F. Tiǧlay and C. Vizman, Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications,, Lett. Math. Phys., 97 (2011), 45.  doi: 10.1007/s11005-011-0464-2.  Google Scholar

[40]

H. Triebel, Theory of Function Spaces,, Birkhäuser Boston, (1983).  doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[41]

M. Wunsch, On the geodesic flow on the group of diffeomorphisms of the circle with a fractional Sobolev right-invariant metric,, J. Nonlinear Math. Phys., 17 (2010), 7.  doi: 10.1142/S1402925110000544.  Google Scholar

[42]

Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272.  doi: 10.1137/S0036141003425672.  Google Scholar

[1]

Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013
[2]

Hicham Zmarrou, Ale Jan Homburg. Dynamics and bifurcations of random circle diffeomorphism. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 719-731. doi: 10.3934/dcdsb.2008.10.719
[3]

Flavia Antonacci, Marco Degiovanni. On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 833-842. doi: 10.3934/dcds.2006.15.833
[4]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389
[5]

Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123
[6]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365
[7]

Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125
[8]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[9]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
[10]

Mohammad Parto-haghighi, Mustafa Inc, Mustafa Bayram, Dumitru Baleanu. Solving the time-fractional Schrodinger equation with the group preserving scheme. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020146
[11]

Nguyen Huy Tuan, Donal O'Regan, Tran Bao Ngoc. Continuity with respect to fractional order of the time fractional diffusion-wave equation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020033
[12]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241
[13]

Yury Neretin. The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions. Journal of Geometric Mechanics, 2017, 9 (2) : 207-225. doi: 10.3934/jgm.2017009
[14]

Carsten Burstedde. On the numerical evaluation of fractional Sobolev norms. Communications on Pure & Applied Analysis, 2007, 6 (3) : 587-605. doi: 10.3934/cpaa.2007.6.587
[15]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[16]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[17]

Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120
[18]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[19]

Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313
[20]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences