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May  2020, 19(5): 2797-2818. doi: 10.3934/cpaa.2020122

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

Panayotis Smyrnelis

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.

Keywords: heteroclinic orbits, Hilbert spaces, bistable potentials, heteroclinic double layers, second and fourth order systems of PDE.
Mathematics Subject Classification: Primary: 34G20, 35J20, 35J35; Secondary: 35B40.
Citation: Panayotis Smyrnelis. Connecting orbits in Hilbert spaces and applications to P.D.E. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2797-2818. doi: 10.3934/cpaa.2020122
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.

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