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Connecting orbits in Hilbert spaces and applications to P.D.E
Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland |
We prove a general theorem on the existence of heteroclinic orbits in Hilbert spaces, and present a method to reduce the solutions of some P.D.E. problems to such orbits. In our first application, we give a new proof in a slightly more general setting of the heteroclinic double layers (initially constructed by Schatzman [
References:
[1] |
S. Alama, L. Bronsard and C. Gui,
Stationary layered solutions in $ \mathbb{R}^2$ for an Allen-Cahn system with multiple well potential, Calc. Var., 5 (1997), 359-390.
doi: 10.1007/s005260050071. |
[2] |
F. Alessio,
Stationary layered solutions for a system of Allen-Cahn type equations, Indiana Univ. Math. J., 62 (2013), 1535-1564.
doi: 10.1512/iumj.2013.62.5108. |
[3] |
F. Alessio and P. Montecchiari,
Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential, J. Fixed Point Theory Appl., 19 (2017), 691-717.
doi: 10.1007/s11784-016-0370-4. |
[4] |
F. Alessio, P. Montecchiari and A. Zuniga,
Prescribed energy connecting orbits for gradient systems, Discrete Contin. Dyn. Syst., 39 (2019), 4895-4928.
doi: 10.3934/dcds.2019200. |
[5] |
N. D. Alikakos and G. Fusco,
On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906.
doi: 10.1512/iumj.2008.57.3181. |
[6] |
N. D. Alikakos and G. Fusco,
Density estimates for vector minimizers and applications, Discrete Contin. Dyn. Syst., 35 (2015), 5631-5663.
doi: 10.3934/dcds.2015.35.5631. |
[7] |
P. Antonopoulos and P. Smyrnelis,
On minimizers of the Hamiltonian system $u'' = \nabla W(u)$, and on the existence of heteroclinic, homoclinic and periodic orbits, Indiana Univ. Math. J., 65 (2016), 1503-1524.
doi: 10.1512/iumj.2016.65.5879. |
[8] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Notas de Matemática, Vol. 50, North-Holland Publishing Company, 1973. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag, New York, 2011. |
[10] |
L. Caffarelli and A. Córdoba,
Uniform convergence of a singular perturbation problem, Commun. Pure Appl. Math., 48 (1995), 1-12.
doi: 10.1002/cpa.3160480101. |
[11] |
T. Cazenave, and A. Haraux, An Introduction to Semilinaer Evolution Equations, Oxford Lecture Series in Mathematics and Its Applications, Clarendon Press, 1998. |
[12] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, 2$^nd$ edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[13] |
G. Fusco,
Layered solutions to the vector Allen-Cahn equation in $ \mathbb{R}^2$. Minimizers and heteroclinic connections, Commun. Pure Appl. Anal., 16 (2017), 1807-1841.
doi: 10.3934/cpaa.2017088. |
[14] |
G. Fusco, G. F. Gronchi and M. Novaga,
On the existence of heteroclinic connections, Sao Paulo J. Math. Sci., 12 (2017), 1-14.
doi: 10.1007/s40863-017-0080-x. |
[15] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, CRC Press, 2006.
![]() ![]() |
[16] |
D. Gilbarg, and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften, Vol. 224, Revised 2$^nd$ edition, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-61798-0. |
[17] |
M. Kreuter, Sobolev Spaces of Vector-valued Functions, Master thesis, Ulm University, Faculty of Mathematics and Economics, 2015. |
[18] |
A. Monteil, and F. Santambrogio, Metric methods for heteroclinic connections in infinite dimensional spaces, preprint, arXiv: 1709.02117.
doi: 10.1002/mma.4072. |
[19] |
O. Savin, Minimal Surfaces and Minimizers of the Ginzburg-Landau energy, in Contemporary Mathematics, Vol. 528, American Mathematical Society, Providence, RI, (2010), 43–57.
doi: 10.1090/conm/528/10413. |
[20] |
M. Schatzman, Asymmetric heteroclinic double layers, Control Optim. Calc. Var. (A tribute to J. L. Lions), 8 (2002), 965–1005.
doi: 10.1051/cocv:2002039. |
[21] |
P. Smyrnelis,
Minimal heteroclinics for a class of fourth order O. D. E. systems, Nonlinear Anal.-Theory Methods Appl., 173 (2018), 154-163.
doi: 10.1016/j.na.2018.04.003. |
[22] |
P. Sternberg,
The effect of a singular perturbation on nonconvex variational problems, Arch. Ration. Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[23] |
P. Sternberg and A. Zuniga,
On the heteroclinic connection problem for multi-well gradient systems, J. of Differ. Equ., 261 (2016), 3987-4007.
doi: 10.1016/j.jde.2016.06.010. |
show all references
References:
[1] |
S. Alama, L. Bronsard and C. Gui,
Stationary layered solutions in $ \mathbb{R}^2$ for an Allen-Cahn system with multiple well potential, Calc. Var., 5 (1997), 359-390.
doi: 10.1007/s005260050071. |
[2] |
F. Alessio,
Stationary layered solutions for a system of Allen-Cahn type equations, Indiana Univ. Math. J., 62 (2013), 1535-1564.
doi: 10.1512/iumj.2013.62.5108. |
[3] |
F. Alessio and P. Montecchiari,
Brake orbit solutions for semilinear elliptic systems with asymmetric double well potential, J. Fixed Point Theory Appl., 19 (2017), 691-717.
doi: 10.1007/s11784-016-0370-4. |
[4] |
F. Alessio, P. Montecchiari and A. Zuniga,
Prescribed energy connecting orbits for gradient systems, Discrete Contin. Dyn. Syst., 39 (2019), 4895-4928.
doi: 10.3934/dcds.2019200. |
[5] |
N. D. Alikakos and G. Fusco,
On the connection problem for potentials with several global minima, Indiana Univ. Math. J., 57 (2008), 1871-1906.
doi: 10.1512/iumj.2008.57.3181. |
[6] |
N. D. Alikakos and G. Fusco,
Density estimates for vector minimizers and applications, Discrete Contin. Dyn. Syst., 35 (2015), 5631-5663.
doi: 10.3934/dcds.2015.35.5631. |
[7] |
P. Antonopoulos and P. Smyrnelis,
On minimizers of the Hamiltonian system $u'' = \nabla W(u)$, and on the existence of heteroclinic, homoclinic and periodic orbits, Indiana Univ. Math. J., 65 (2016), 1503-1524.
doi: 10.1512/iumj.2016.65.5879. |
[8] |
H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Notas de Matemática, Vol. 50, North-Holland Publishing Company, 1973. |
[9] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag, New York, 2011. |
[10] |
L. Caffarelli and A. Córdoba,
Uniform convergence of a singular perturbation problem, Commun. Pure Appl. Math., 48 (1995), 1-12.
doi: 10.1002/cpa.3160480101. |
[11] |
T. Cazenave, and A. Haraux, An Introduction to Semilinaer Evolution Equations, Oxford Lecture Series in Mathematics and Its Applications, Clarendon Press, 1998. |
[12] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, 2$^nd$ edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[13] |
G. Fusco,
Layered solutions to the vector Allen-Cahn equation in $ \mathbb{R}^2$. Minimizers and heteroclinic connections, Commun. Pure Appl. Anal., 16 (2017), 1807-1841.
doi: 10.3934/cpaa.2017088. |
[14] |
G. Fusco, G. F. Gronchi and M. Novaga,
On the existence of heteroclinic connections, Sao Paulo J. Math. Sci., 12 (2017), 1-14.
doi: 10.1007/s40863-017-0080-x. |
[15] |
L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, CRC Press, 2006.
![]() ![]() |
[16] |
D. Gilbarg, and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der mathematischen Wissenschaften, Vol. 224, Revised 2$^nd$ edition, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-642-61798-0. |
[17] |
M. Kreuter, Sobolev Spaces of Vector-valued Functions, Master thesis, Ulm University, Faculty of Mathematics and Economics, 2015. |
[18] |
A. Monteil, and F. Santambrogio, Metric methods for heteroclinic connections in infinite dimensional spaces, preprint, arXiv: 1709.02117.
doi: 10.1002/mma.4072. |
[19] |
O. Savin, Minimal Surfaces and Minimizers of the Ginzburg-Landau energy, in Contemporary Mathematics, Vol. 528, American Mathematical Society, Providence, RI, (2010), 43–57.
doi: 10.1090/conm/528/10413. |
[20] |
M. Schatzman, Asymmetric heteroclinic double layers, Control Optim. Calc. Var. (A tribute to J. L. Lions), 8 (2002), 965–1005.
doi: 10.1051/cocv:2002039. |
[21] |
P. Smyrnelis,
Minimal heteroclinics for a class of fourth order O. D. E. systems, Nonlinear Anal.-Theory Methods Appl., 173 (2018), 154-163.
doi: 10.1016/j.na.2018.04.003. |
[22] |
P. Sternberg,
The effect of a singular perturbation on nonconvex variational problems, Arch. Ration. Mech. Anal., 101 (1988), 209-260.
doi: 10.1007/BF00253122. |
[23] |
P. Sternberg and A. Zuniga,
On the heteroclinic connection problem for multi-well gradient systems, J. of Differ. Equ., 261 (2016), 3987-4007.
doi: 10.1016/j.jde.2016.06.010. |
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