Homogenization of an eigenvalue problem in a two-component domain with an interfacial barrier / Eleanor Bañanola Gemida

Thesis (Master of Science in Mathematics) -- University of the Philippines Los Baños, June 2018.

The study deals with the homogenization of a stationary elliptic eigenvalue problem with oscillating coefficients in a domain n C !RN which is the union of two subdomains ni and 02, separated by an interface T-. The component l5 is the union of the disjoint e-periodic translated sets sl, where Y lies in the reference cell Y. On the other hand, the component f2t is connected and defined as 0\0,. Mathematically, study the asymptotic behaviour as E ➔ 0 of the problem-div(A-Vu,) = Nu-div(AV,) = Nu5

A-Vu~nf, =-AVu5n5
A-Vu;nf, =-eh(f- u5)
uf= 0
in Di,
in 05
on re,
on f",
on 8D,
where € R and n; is the unitary outward normal to n:, i = 1, 2. (1)

The main goal is to analyze the convergence of the eigenvalues and eigenvectors of the heat; equation described in (1). We obtain characterizations of the eigenvalues
and give homogenization results for the case. I using the periodic unfolding method. For <I, the " eigenvalue of (1) converges to the €11 ' eigenvalue of the limit problem, for the whole sequence. The same convergence result is obtained for the corresponding eigenvectors, for a subsequence only. The convergence for the whole sequence is achieved when the associated eigenvalue is simple. For the Case ) = l, we only have convergence results up to a subsequence