• Previous Article
    Ground state solution and multiple solutions to elliptic equations with exponential growth and singular term
  • CPAA Home
  • This Issue
  • Next Article
    $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients
May  2020, 19(5): 2797-2818. doi: 10.3934/cpaa.2020122

Connecting orbits in Hilbert spaces and applications to P.D.E

Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland

Received  May 2019 Revised  October 2019 Published  March 2020

Fund Project: The author was partially supported by the National Science Centre, Poland (Grant No. 2017/26/E/ST1/00817)

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 [20]), since this result is particularly relevant for phase transition systems. In our second application, we obtain a solution of a fouth order P.D.E. satisfying similar boundary conditions.

Citation: Panayotis Smyrnelis. Connecting orbits in Hilbert spaces and applications to P.D.E. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2797-2818. doi: 10.3934/cpaa.2020122
References:
[1]

S. AlamaL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. AlessioP. Montecchiari and A. Zuniga, Prescribed energy connecting orbits for gradient systems, Discrete Contin. Dyn. Syst., 39 (2019), 4895-4928.  doi: 10.3934/dcds.2019200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag, New York, 2011.  Google Scholar

[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.  Google Scholar

[11]

T. Cazenave, and A. Haraux, An Introduction to Semilinaer Evolution Equations, Oxford Lecture Series in Mathematics and Its Applications, Clarendon Press, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. FuscoG. 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.  Google Scholar

[15] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, CRC Press, 2006.   Google Scholar
[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.  Google Scholar

[17]

M. Kreuter, Sobolev Spaces of Vector-valued Functions, Master thesis, Ulm University, Faculty of Mathematics and Economics, 2015. Google Scholar

[18]

A. Monteil, and F. Santambrogio, Metric methods for heteroclinic connections in infinite dimensional spaces, preprint, arXiv: 1709.02117. doi: 10.1002/mma.4072.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. AlamaL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

F. AlessioP. Montecchiari and A. Zuniga, Prescribed energy connecting orbits for gradient systems, Discrete Contin. Dyn. Syst., 39 (2019), 4895-4928.  doi: 10.3934/dcds.2019200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag, New York, 2011.  Google Scholar

[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.  Google Scholar

[11]

T. Cazenave, and A. Haraux, An Introduction to Semilinaer Evolution Equations, Oxford Lecture Series in Mathematics and Its Applications, Clarendon Press, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. FuscoG. 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.  Google Scholar

[15] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, CRC Press, 2006.   Google Scholar
[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.  Google Scholar

[17]

M. Kreuter, Sobolev Spaces of Vector-valued Functions, Master thesis, Ulm University, Faculty of Mathematics and Economics, 2015. Google Scholar

[18]

A. Monteil, and F. Santambrogio, Metric methods for heteroclinic connections in infinite dimensional spaces, preprint, arXiv: 1709.02117. doi: 10.1002/mma.4072.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The sequence $ -\infty = x_0<y_1<x_1\leq y_2<x_2<\ldots<x_{2N} = \infty $, ($ N = 2 $)
[1]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[2]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343

[3]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967

[4]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[5]

E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261

[6]

François Hamel, Jean-Michel Roquejoffre. Heteroclinic connections for multidimensional bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 101-123. doi: 10.3934/dcdss.2011.4.101

[7]

Paul H. Rabinowitz. A new variational characterization of spatially heteroclinic solutions of a semilinear elliptic PDE. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 507-515. doi: 10.3934/dcds.2004.10.507

[8]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071

[9]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69

[10]

Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039

[11]

Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations & Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009

[12]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778

[13]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[14]

Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769

[15]

Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569

[16]

Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2825-2851. doi: 10.3934/dcds.2012.32.2825

[17]

Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 255-271. doi: 10.3934/dcdsb.2011.15.255

[18]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020176

[19]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005

[20]

Lorenzo Arona, Josep J. Masdemont. Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem. Conference Publications, 2007, 2007 (Special) : 64-74. doi: 10.3934/proc.2007.2007.64

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (62)
  • HTML views (77)
  • Cited by (0)

Other articles
by authors

[Back to Top]