2018/05/29 by Fusco, Giorgio, Gronchi, Giovanni F., Novaga, Matteo
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1805.11563
We consider a potential W:Rm→ R with two different global minima a-, a+ and, under a symmetry assumption, we use a variational approach to show that the Hamiltonian system u=Wu(u), \hskip 2cm (1) has a family of T-periodic solutions uT which, along a sequence Tj→+∞, converges locally to a heteroclinic solution that connects a- to a+. We then focus on the elliptic system Δu=Wu(u), u:R2→ Rm, \hskip 2cm (2) that we interpret as an infinite dimensional analogous of (1), where x plays the role of time and W is replaced by the action functional JR(u)=∫R((1)/(2)\vert uy\vert2+W(u))dy. We assume that JR has two different global minimizers u-, u+:R→ Rm in the set of maps that connect a- to a+. We work in a symmetric context and prove, via a minimization procedure, that (2) has a family of solutions uL:R2→ Rm, which is L-periodic in x, converges to a_± as y→±∞ and, along a sequence Lj→+∞, converges locally to a heteroclinic solution that connects u- to u+.