Skip to main content Accessibility help
×
Home

A tale of two approaches to heteroclinic solutions for Φ-Laplacian systems

  • Yuan L. Ruan (a1)

Abstract

In this article, the existence of heteroclinic solution of a class of generalized Hamiltonian system with potential $V : {\open R}^{n} \longmapsto {\open R}$ having a finite or infinite number of global minima is studied. Examples include systems involving the p-Laplacian operator, the curvature operator and the relativistic operator. Generalized conservation of energy is established, which leads to the property of equipartition of energy enjoyed by heteroclinic solutions. The existence problem of heteroclinic solution is studied using both variational method and the metric method. The variational approach is classical, while the metric method represents a more geometrical point of view where the existence problem of heteroclinic solution is reduced to that of geodesic in a proper length metric space. Regularities of the heteroclinic solutions are discussed. The results here not only provide alternative solution methods for Φ-Laplacian systems, but also improve existing results for the classical Hamiltonian system. In particular, the conditions imposed upon the potential are very mild and new proof for the compactness is given. Finally in ℝ2, heteroclinic solutions are explicitly written down in closed form by using complex function theory.

Copyright

References

Hide All
1Alama, S., Bronsard, L. and Gui, C.. Stationary layered solutions in R 2 for an Allen-Cahn system with multiple well potential. Calc. Var. Partial Diff. Equ. 5 (1997), 359390.
2Alikakos, N.. A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system δuw u(u) = 0. Comm. Partial Diff. Equ. 37 (2012), 20932115.
3Alikakos, N. and Fusco, G.. On the connection problem for potentials with several global minima. Indiana Univ. Math. J. 57 (2008), 18711906.
4Alikakos, N., Betelú, S. and Chen, X.. Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities. Eur. J. Appl. Math. 17 (2006), 525556.
5Ambrosio, L. and Tilli, P.. Topics on analysis in metric spaces , vol. 25 (Oxford: Oxford University Press, 2004).
6Arcoya, D., Bereanu, C. and Torres, P.. Critical point theory for the Lorentz force equation. Arch. Ration. Mech. Anal. 232 (2019), 16851724.
7Arendt, W. and Kreuter, M.. Mapping theorems for sobolev spaces of vector-valued functions. Studia Math. 240 (2018), 275299.
8Bereanu, C. and Mawhin, J.. Existence and multiplicity results for some nonlinear problems with singular ϕ-laplacian. J. Diff. Equ. 243 (2007), 536557.
9Bethuel, F., Orlandi, G. and Smets, D.. Slow motion for gradient systems with equal depth multiple-well potentials. J. Diff. Equ. 250 (2011), 5394.
10Bianconi, B. and Papalini, F.. Non-autonomous boundary value problems on the real line. Discrete Contin. Dyn. Syst. 15 (2006), 759776.
11Bolotin, S. V. and Kozlov, V. V.. On asymptotic solutions of equations of dynamics. Vestnik Moskov. Univ. Ser. I., Matem. Mekh. 4 (1980), 8489.
12Bonheure, D., Obersnel, F., Omari, P., et al. Heteroclinic solutions of the prescribed curvature equation with a double-well potential. Diff. Int. Equ. 26 (2013), 14111428.
13Bonheure, D., Coelho, I. and Nys, M.. Heteroclinic solutions of singular quasilinear bistable equations. NoDEA Nonlinear Diff. Equ. Appl. 24 (2017), 2.
14Bridson, M. and Haefliger, A.. Metric spaces of non-positive curvature, vol. 319 (Berlin-Heidelberg: Springer-Verlag, 2013).
15Burago, D., Burago, Y., Burago, I. D. and Ivanov, S.. A course in metric geometry, vol. 33 (Rhode Island: American Mathematical Soc., 2001).
16Byeon, J., Montecchiari, P. and Rabinowitz, P. H.. A double well potential system. Anal. PDE 9 (2016), 17371772.
17Cabada, A. and Cid, J.. Heteroclinic solutions for non-autonomous boundary value problems with singular ϕ-laplacian operators. Discrete Contin. Dyn. Syst 2009 (2009), 118122.
18Chang, K. C.. Lecture notes on calculus of variations, vol. 6 (Singapore: World Scientific, 2016).
19Cianchi, A. and Maz'ya, V.. Quasilinear elliptic problems with general growth and merely integrable, or measure, data. Nonlinear Anal. 164 (2017), 189215.
20Dacorogna, B.. Direct methods in the calculus of variations, vol. 78 (Berlin-Heidelberg: Springer-Verlag, 2007).
21Felmer, P. L.. Heteroclinic orbits for spatially periodic hamiltonian systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 477497.
22Heinonen, J., Koskela, P., Shanmugalingam, N. and Tyson, J.. Sobolev spaces on metric measure spaces, vol. 27 (Cambridge: Cambridge University Press, 2015).
23Kozlov, V. V.. Calculus of variations in the large and classical mechanics. Russian Math. Surv. 40 (1985), 3360.
24Leoni, G.. A first course in Sobolev spaces, 2nd edn (Rhode Island: American Mathematical Soc., 2017).
25Manasevich, R. and Mawhin, J.. Boundary value problems for nonlinear perturbations of vector p-laplacian-like operators. J. Korean Math. Soc. 37 (2000), 665685.
26Marcus, M. and Mizel, V.. Absolute continuity on tracks and mappings of Sobolev spaces. Arch. Ration. Mech. Anal. 45 (1972), 294320.
27Monteil, A. and Santambrogio, F.. Metric methods for heteroclinic connections in infinite dimensional spaces. arXiv preprint arXiv:1709.02117 (2017).
28Monteil, A. and Santambrogio, F.. Metric methods for heteroclinic connections. Math. Meth. Appl. Sci. 41 (2018), 10191024.
29Rabinowitz, P. H.. Periodic and heteroclinic orbits for a periodic Hamiltonian system. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), 331346.
30Rabinowitz, P. H.. Homoclinic and heteroclinic orbits for a class of hamiltonian systems. Calc. Var. Partial Diff. Equ. 1 (1993), 136.
31Rabinowitz, P. H.. On a theorem of strobel. Calc. Var. Partial Diff. Equ. 12 (2001), 399415.
32Rabinowitz, P. H. and Tanaka, K.. Some results on connecting orbits for a class of hamiltonian systems. Math. Z. 206 (1991), 473499.
33Sourdis, C.. The heteroclinic connection problem for general double-well potentials. Mediterr. J. Math. 13 (2016), 46934710.
34Sternberg, P.. The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101 (1988), 209260.
35Sternberg, P.. Vector-valued local minimizers of nonconvex variational problems. Rocky Mountain J. Math. 21 (1991), 799807.
36Strobel, K.. Multi-bump solutions for a class of periodic hamiltonian systems. Thesis (Wisconsin–Madison: University of Wisconsin, 1995).
37Zuniga, A. and Sternberg, P.. On the heteroclinic connection problem for multi-well gradient systems. J. Diff. Equ. 261 (2016), 39874007.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed