An Infinite-Horizon Variational Problem on an Infinite Strip

This article is devoted to the study of a variational problem on an infinite strip Omega = (0, infinity) x (0, L). It generalizes previous works which dealt with the one-dimensional case, notably the one by Leizarowitz and Mizel. More precisely, given g is an element of H-3/2 (partial derivative Omega) x H-1/2 (partial derivative Omega) such that (1) g = 0 on (0, infinity) x {0} U (0, infinity) x {L}, we seek a "minimal solution" for the functional I [u] = integral(f)(Omega) (u, Du, D(2)u) dx, (2) for u is an element of A(g) {v is an element of H-loc(2) (Omega) : (upsilon vertical bar partial derivative Omega, partial derivative upsilon/partial derivative v vertical bar sigma Omega) = g} partial derivative upsilon/partial derivative v where partial derivative upsilon/partial derivative v vertical bar partial derivative Omega is the outward normal derivative on partial derivative Omega, for a free energy integrand f satisfying some natural assumptions. Since the infimum of I [.] on A(g) is typically either +infinity or -infinity, we consider the expression J Omega(k) [u] = 1/vertical bar Omega(K vertical bar) integral(Omega k) f(u, Du, D(2)u) dx, where Omega(k) = (0, k) x (0, L), for any k > 0, and study the limit as k tends to infinity. As k -> infinity, the limit of J(Omega K) [u] represents the average energy of u on Omega, and whenever this limit has meaning we define (3) J[u] = lim inf J(Omega k) [u]. k ->infinity Our main result establishes, for any g satisfying (1), the existence of a minimal solution u for (2), i.e., u is a minimizer for J [.] and for each k > 0 it is a minimizer for J(Omega k) [.] among all functions satisfying (upsilon vertical bar partial derivative Omega(k,) partial derivative upsilon/partial derivative v vertical bar partial derivative Omega(k)) = (u vertical bar partial derivative Omega(k,) partial derivative upsilon/partial derivative v vertical bar partial derivative Omega(k)).