Periodic homogenization of Dirichlet problem for divergence type elliptic operators
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Date
26/11/2015Author
Aleksanyan, Hayk
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Abstract
The thesis studies homogenization of Dirichlet boundary value problems for divergence
type elliptic operators, and the associated boundary layer issues. This
type of problems for operators with periodically oscillating coeffcients, and fixed
boundary data are by now a classical topic largely due to the celebrated work by
Avellaneda and Lin from late 80's. The case when the operator and the Dirichlet
boundary data exhibit periodic oscillations simultaneously was a longstanding
open problem, and a progress in this direction has been achieved only very recently,
in 2012, by Gerard-Varet and Masmoudi who proved a homogenization
result for the simultaneously oscillating case with an algebraic rate of convergence
in L2.
Aimed at understanding the homogenization process of oscillating boundary
data, in the first part of the thesis we introduce and develop Fourier-analytic
ideas into the study of homogenization of Dirichlet boundary value problems for
elliptic operators in divergence form. In smooth and bounded domains, for fixed
operator and periodically oscillating boundary data we prove pointwise, as well
as Lp convergence results the homogenization problem. We then investigate the
optimality (sharpness) of our Lp upper bounds. Next, for the above mentioned
simultaneously oscillating problem studied by Gerard-Varet and Masmoudi, we
establish optimal Lp bounds for homogenization in some class of operators.
For domains with non smooth boundary, we study similar boundary value
homogenization problems for scalar equations set in convex polygonal domains.
In the vein of smooth boundaries, here as well for problems with fixed operator
and oscillating Dirichlet data we prove pointwise, and Lp convergence results, and
study the optimality of our Lp bounds. Although the statements are somewhat
similar with the smooth setting, challenges for this case are completely different
due to a radical change in the geometry of the domain.
The second part of the work is concerned with the analysis of boundary layers
arising in periodic homogenization. A key difficulty toward the homogenization
of Dirichlet problem for elliptic systems in divergence form with periodically oscillating
coefficients and boundary condition lies in identification of the limiting
Dirichlet data corresponding to the effective problem. This question has been addressed
in the aforementioned work by Gerard-Varet and Masmoudi on the way of
proving their main homogenization result. Despite the progress in this direction,
some very basic questions remain unanswered, for instance the regularity of this
effective data on the boundary. This issue is directly linked with the up to the
boundary regularity of homogenized solutions, but perhaps more importantly has
a potential to cast light on the homogenization process. We initiate the study of
this regularity problem, and prove certain Lipschitz continuity result. The work
also comprises a study on asymptotic behaviour of solutions to boundary layer
systems set in halfspaces. By a new construction we show that depending on the
normal direction of the hyperplane, convergence of the solutions toward their tails
far away from the boundaries can be arbitrarily slow. This last result, combined
with the previous studies gives an almost complete picture of the situation.