# American Institute of Mathematical Sciences

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 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.

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##### 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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