טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentYudovich Ilya
SubjectTwo Dimensional Unbounded-domain Variational Problem
DepartmentDepartment of Applied Mathematics
Supervisor Professor Itai Shafrir


Abstract

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                gϵH3/2 (∂Ω)?H1/2 (∂Ω) such that

  1.       g = 0       on      (0,∞)?{0}∩(0,∞)?{L}.
                We seek a “minimal solution” for the functional

  2.                              I[u] = ∫Ω   f(u,Du,D2udx,
  3. for           uϵAg≔{vvϵH2loc(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 Ag is typically either ∞ or -∞, we consider the expression
            JΩk[u]=|k|-1Ωk   f(u,Du,D2udx,
            where k = (0,k) ?(0,L), for any > 0, and study the limit as k tends to infinity.
            As k→∞, the limit of JΩk[u] represents the average energy of u on Ω, and whenever this limit has meaning we define

  4.        J[u]=liminf k→∞ JΩk[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).