November  2019, 12(7): 1867-1877. doi: 10.3934/dcdss.2019123

The Morse property for functions of Kirchhoff-Routh path type

1. 

Mathematisches Institut, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany

2. 

Dipartimento di Matematica, Università di Pisa, Via Bonanno 25B, 56126 Pisa, Italy

3. 

Dipartimento SBAI, Università di Roma "La Sapienza", Via Antonio Scarpa 16, 00161 Roma, Italy

* Corresponding author: Thomas Bartsch

Dedicated to Norman Dancer, with friendship and esteem

Received  August 2017 Revised  February 2018 Published  December 2018

Fund Project: The first author is supported by funds "Agreement between Sapienza University of Roma and University of Giessen".

For a bounded domain
$ \Omega\subset\mathbb{R}^n $
let
$ H_\Omega:\Omega\times\Omega\to\mathbb{R} $
be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary
$ {\mathcal C}^2 $
function
$ f:{\mathcal D}\to\mathbb{R} $
, defined on an open subset
$ {\mathcal D}\subset\mathbb{R}^{nN} $
, and fixed coefficients
$ \lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\} $
we consider the function
$ f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R} $
defined as
$ f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum\limits_{j,k = 1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k). $
We prove that
$ f_\Omega $
is a Morse function for most domains
$ \Omega $
of class
$ {\mathcal C}^{m+2,\alpha} $
, any
$ m\ge0 $
,
$ 0<\alpha<1 $
. This applies in particular to the Robin function
$ h:\Omega\to\mathbb{R} $
,
$ h(x) = H_\Omega(x,x) $
, and to the Kirchhoff-Routh path function where
$ \Omega\subset\mathbb{R}^2 $
,
$ {\mathcal D} = \{x\in\mathbb{R}^{2N}: x_j\ne x_k \; \text{for }\; j\ne k \} $
, and
$ f(x_1,\dots,x_N) = - \frac{1}{2\pi}\sum\limits_{{j,k = 1}\atop{j\ne k}}^N\lambda_j\lambda_k\log|x_j-x_k|. $
Citation: Thomas Bartsch, Anna Maria Micheletti, Angela Pistoia. The Morse property for functions of Kirchhoff-Routh path type. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1867-1877. doi: 10.3934/dcdss.2019123
References:
[1]

T. Bartsch, Periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Arch. Math., 107 (2016), 413-422.  doi: 10.1007/s00013-016-0928-9.  Google Scholar

[2]

T. Bartsch and Q. Dai, Periodic solutions of the $N$-vortex Hamiltonian in planar domains, Diff. Eq., 260 (2016), 2275-2295.  doi: 10.1016/j.jde.2015.10.002.  Google Scholar

[3]

T. Bartsch and B. Gebhard, Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Math. Ann., 369 (2017), 627-651.  doi: 10.1007/s00208-016-1505-z.  Google Scholar

[4]

T. BartschT. D'Aprile and A. Pistoia, Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1027-1047.  doi: 10.1016/j.anihpc.2013.01.001.  Google Scholar

[5]

T. BartschA. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Diff. Equ., 26 (2006), 265-282.  doi: 10.1007/s00526-006-0004-6.  Google Scholar

[6]

T. Bartsch and A. Pistoia, Critical points of the $N$-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75 (2015), 726-744.  doi: 10.1137/140981253.  Google Scholar

[7]

T. BartschA. Pistoia and T. Weth, N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 297 (2010), 653-686.  doi: 10.1007/s00220-010-1053-4.  Google Scholar

[8]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhauser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[9]

D. Cao, Z. Liu and J. Wei, Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212 (2014), 179-217. doi: 10.1007/s00205-013-0692-y.  Google Scholar

[10]

M. del PinoM. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Part. Diff. Equ., 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5.  Google Scholar

[11]

P. EspositoM. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[12]

A. Fonda, M. Garrione and P. Gidoni, Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal., 5 (2016), 367-382. doi: 10.1515/anona-2015-0122.  Google Scholar

[13]

B. Gebhard, Periodic solutions for the N-vortex problem via a superposition principle, preprint, arXiv: 1708.08888 Google Scholar

[14]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin New York, 1977.  Google Scholar

[15]

J. Hadamard, Mémoire sur le probleme d'analyse relatif a l'equilibre des plaques elastiques encastrees, Mémoires présentés par divers savants a l'Académie des Sciences, 1908. Google Scholar

[16]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Mathematical Society Lecture Note Series, 318. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511546730.  Google Scholar

[17]

G. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876. Google Scholar

[18]

C. Kuhl, Symmetric equilibria for the N-vortex-problem, J. Fixed Point Theory Appl., 17 (2015), 597-624.  doi: 10.1007/s11784-015-0242-3.  Google Scholar

[19]

C. Kuhl, Equilibria for the N-vortex-problem in a general bounded domain, J. Math. Anal. Appl., 433 (2016), 1531-1560. doi: 10.1016/j.jmaa.2015.08.055.  Google Scholar

[20]

C. C. Lin, On the motion of vortices in 2D Ⅰ. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sc., 27 (1941), 570-575.  doi: 10.1073/pnas.27.12.570.  Google Scholar

[21]

C. C. Lin, On the motion of vortices in 2D Ⅱ. Some further properties on the Kirchhoff-Routh function, Proc. Nat. Acad. Sc., 27 (1941), 575-577. doi: 10.1073/pnas.27.12.575.  Google Scholar

[22]

F. H. Lin and T.-C. Lin, Minimax solutions of the Ginzburg-Landau equations, Selecta Math., 3 (1997), 99-113.  doi: 10.1007/s000290050007.  Google Scholar

[23]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied mathematical sciences, 96, Springer, New York, 1994. doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[24]

A. M. Micheletti and A. Pistoia, Non degeneracy of critical points of the Robin function with respect to deformations of the domain, Potential Anal., 40 (2014), 103-116.  doi: 10.1007/s11118-013-9340-2.  Google Scholar

[25]

P. K. Newton, The $N$-vortex Problem, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[26]

E. J. Routh, Some applications of conjugate functions, Proc. London Math. Soc., 12 (1881), 73-89.  doi: 10.1112/plms/s1-12.1.73.  Google Scholar

[27]

D. Smets and J. Van Schaftingen, Desingularization of vortices for the Euler equation, Arch. Rational Mech. Anal., 198 (2010), 869-925.  doi: 10.1007/s00205-010-0293-y.  Google Scholar

show all references

References:
[1]

T. Bartsch, Periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Arch. Math., 107 (2016), 413-422.  doi: 10.1007/s00013-016-0928-9.  Google Scholar

[2]

T. Bartsch and Q. Dai, Periodic solutions of the $N$-vortex Hamiltonian in planar domains, Diff. Eq., 260 (2016), 2275-2295.  doi: 10.1016/j.jde.2015.10.002.  Google Scholar

[3]

T. Bartsch and B. Gebhard, Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Math. Ann., 369 (2017), 627-651.  doi: 10.1007/s00208-016-1505-z.  Google Scholar

[4]

T. BartschT. D'Aprile and A. Pistoia, Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1027-1047.  doi: 10.1016/j.anihpc.2013.01.001.  Google Scholar

[5]

T. BartschA. Micheletti and A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Diff. Equ., 26 (2006), 265-282.  doi: 10.1007/s00526-006-0004-6.  Google Scholar

[6]

T. Bartsch and A. Pistoia, Critical points of the $N$-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75 (2015), 726-744.  doi: 10.1137/140981253.  Google Scholar

[7]

T. BartschA. Pistoia and T. Weth, N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 297 (2010), 653-686.  doi: 10.1007/s00220-010-1053-4.  Google Scholar

[8]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhauser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[9]

D. Cao, Z. Liu and J. Wei, Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212 (2014), 179-217. doi: 10.1007/s00205-013-0692-y.  Google Scholar

[10]

M. del PinoM. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Part. Diff. Equ., 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5.  Google Scholar

[11]

P. EspositoM. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[12]

A. Fonda, M. Garrione and P. Gidoni, Periodic perturbations of Hamiltonian systems, Adv. Nonlinear Anal., 5 (2016), 367-382. doi: 10.1515/anona-2015-0122.  Google Scholar

[13]

B. Gebhard, Periodic solutions for the N-vortex problem via a superposition principle, preprint, arXiv: 1708.08888 Google Scholar

[14]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin New York, 1977.  Google Scholar

[15]

J. Hadamard, Mémoire sur le probleme d'analyse relatif a l'equilibre des plaques elastiques encastrees, Mémoires présentés par divers savants a l'Académie des Sciences, 1908. Google Scholar

[16]

D. Henry, Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Mathematical Society Lecture Note Series, 318. Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511546730.  Google Scholar

[17]

G. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876. Google Scholar

[18]

C. Kuhl, Symmetric equilibria for the N-vortex-problem, J. Fixed Point Theory Appl., 17 (2015), 597-624.  doi: 10.1007/s11784-015-0242-3.  Google Scholar

[19]

C. Kuhl, Equilibria for the N-vortex-problem in a general bounded domain, J. Math. Anal. Appl., 433 (2016), 1531-1560. doi: 10.1016/j.jmaa.2015.08.055.  Google Scholar

[20]

C. C. Lin, On the motion of vortices in 2D Ⅰ. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sc., 27 (1941), 570-575.  doi: 10.1073/pnas.27.12.570.  Google Scholar

[21]

C. C. Lin, On the motion of vortices in 2D Ⅱ. Some further properties on the Kirchhoff-Routh function, Proc. Nat. Acad. Sc., 27 (1941), 575-577. doi: 10.1073/pnas.27.12.575.  Google Scholar

[22]

F. H. Lin and T.-C. Lin, Minimax solutions of the Ginzburg-Landau equations, Selecta Math., 3 (1997), 99-113.  doi: 10.1007/s000290050007.  Google Scholar

[23]

C. Marchioro and M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Applied mathematical sciences, 96, Springer, New York, 1994. doi: 10.1007/978-1-4612-4284-0.  Google Scholar

[24]

A. M. Micheletti and A. Pistoia, Non degeneracy of critical points of the Robin function with respect to deformations of the domain, Potential Anal., 40 (2014), 103-116.  doi: 10.1007/s11118-013-9340-2.  Google Scholar

[25]

P. K. Newton, The $N$-vortex Problem, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[26]

E. J. Routh, Some applications of conjugate functions, Proc. London Math. Soc., 12 (1881), 73-89.  doi: 10.1112/plms/s1-12.1.73.  Google Scholar

[27]

D. Smets and J. Van Schaftingen, Desingularization of vortices for the Euler equation, Arch. Rational Mech. Anal., 198 (2010), 869-925.  doi: 10.1007/s00205-010-0293-y.  Google Scholar

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