The Morse property for functions of Kirchhoff-Routh path type

Thomas Bartsch; Anna Maria Micheletti; Angela Pistoia

doi:10.3934/dcdss.2019123

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

The Morse property for functions of Kirchhoff-Routh path type

Thomas Bartsch ^1,, , Anna Maria Micheletti ^2, and Angela Pistoia ^3,

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|. $

Keywords: Kirchhoff-Routh path function, Morse function, transversality theorem.

Mathematics Subject Classification: Primary: 35J08; Secondary: 35J25, 35Q31, 76B47.

Citation: Thomas Bartsch, Anna Maria Micheletti, Angela Pistoia. The Morse property for functions of Kirchhoff-Routh path type. Discrete and 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.
[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.
[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.
[4]	T. Bartsch, T. 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.
[5]	T. Bartsch, A. 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.
[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.
[7]	T. Bartsch, A. 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.
[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.
[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.
[10]	M. del Pino, M. 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.
[11]	P. Esposito, M. 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.
[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.
[13]	B. Gebhard, Periodic solutions for the N-vortex problem via a superposition principle, preprint, arXiv: 1708.08888
[14]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin New York, 1977.
[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.
[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.
[17]	G. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	P. K. Newton, The $N$-vortex Problem, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-1-4684-9290-3.
[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.
[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.

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.
[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.
[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.
[4]	T. Bartsch, T. 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.
[5]	T. Bartsch, A. 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.
[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.
[7]	T. Bartsch, A. 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.
[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.
[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.
[10]	M. del Pino, M. 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.
[11]	P. Esposito, M. 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.
[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.
[13]	B. Gebhard, Periodic solutions for the N-vortex problem via a superposition principle, preprint, arXiv: 1708.08888
[14]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin New York, 1977.
[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.
[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.
[17]	G. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	P. K. Newton, The $N$-vortex Problem, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-1-4684-9290-3.
[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.
[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.

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