This thesis is devoted to the study of a variational problem on an
infinite strip Ω = (0,∞)?(0,L).
It
generalizes previous works which dealt with the one dimensional case, notably
the one by Leizarowitz and Mizel.
More precisely, given gϵH^{3/2}
(∂Ω)?H^{1/2} (∂Ω) such that

g = 0 on (0,∞)?{0}∩(0,∞)?{L}.
We seek a “minimal solution” for the functional
I[u] = ∫_{Ω }f(u,Du,D^{2}u) dx,

for uϵA_{g}≔{vvϵH^{2}_{loc}(Ω): (v_{∂Ω},
∂v/∂υ_{∂Ω}) = g},
where ∂v/∂υ_{∂Ω} is the outward normal derivative on
∂Ω, for a free energy integrand f satisfying some natural
assumptions.
Since the infimum of I[∙] on A_{g}
is typically either ∞ or ∞, we consider the expression
J_{Ωk}[u]=Ω_{k}^{1}∫_{Ωk }f(u,Du,D^{2}u) dx,
where Ω_{k} = (0,k) ?(0,L), for any k > 0,
and study the limit as k tends to infinity.
As k→∞, the limit of J_{Ωk}[u]
represents the average energy of u on Ω, and whenever this limit
has meaning we define

J[u]=liminf _{k→∞}
J_{Ωk}[u].
Our
main result establishes, for any g satisfying (1),
the existence of minimal
solution u for (2),
i.e.,
u is a minimize for J[∙] among all functions
satisfying (v_{∂Ω},∂v/∂υ_{∂Ω}) =(u_{∂Ω},∂u/∂υ_{∂Ω}).