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On good deformations of $ A_m $-singularities
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 |
$ \Omega\subset\mathbb{R}^n $ |
$ H_\Omega:\Omega\times\Omega\to\mathbb{R} $ |
$ {\mathcal C}^2 $ |
$ f:{\mathcal D}\to\mathbb{R} $ |
$ {\mathcal D}\subset\mathbb{R}^{nN} $ |
$ \lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\} $ |
$ f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R} $ |
$ 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). $ |
$ f_\Omega $ |
$ \Omega $ |
$ {\mathcal C}^{m+2,\alpha} $ |
$ m\ge0 $ |
$ 0<\alpha<1 $ |
$ h:\Omega\to\mathbb{R} $ |
$ h(x) = H_\Omega(x,x) $ |
$ \Omega\subset\mathbb{R}^2 $ |
$ {\mathcal D} = \{x\in\mathbb{R}^{2N}: x_j\ne x_k \; \text{for }\; j\ne k \} $ |
$ 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|. $ |
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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