Abstract
In this paper we study the asymptotic behavior of a Ginzburg-Landau problem in a ε-periodically perforated domain of R n with mixed Dirichlet-Neumann conditions. The holes can verify two different situations. In the first one they have size ε and a homogeneous Dirichlet condition is posed on a flat portion of each hole, whose size is an order smaller than ε, the Neumann condition being posed on the remaining part. In the second situation, we consider two kinds of ε-periodic holes, one of size of order smaller than ε, where a homogeneous Dirichlet condition is prescribed and the other one of size ε, on which a non-homogeneous Neumann condition is given. Moreover, in this case as ε goes to zero, the two families of holes approach each other. In both situations a homogeneous Dirichlet condition is also prescribed on the whole exterior boundary of the domain.MSC: 35J20, 35J25, 35B25, 35J55, 35B40.
Highlights
Let be a bounded set in Rn, with Lipschitz boundary ∂ and Y be [– ]n.We consider two kinds of holes removed from, both periodically distributed
In this paper we study the homogenization process of the following vectorial nonlinear problem with mixed boundary conditions:
By using the energy method, for n = and n = we prove the weakly convergence in H ( ) of a suitable extension of the sequence of solutions uτε,σ of ( . ) to a function uτ,σ, unique solution of the following limit problem:
Summary
The first kind is of size ε, ε being a positive parameter It is obtained by rescaling a reference hole Q (a cube or a smoothed one) contained in Y and in the half plane {x ≥ }, a piece of which is on the hyperplane. (c) the capacity in R of a set obtained by doubling the Dirichlet reference hole by reflection with respect to the hyperplane {x = } if σ >. Many authors (see for example [ – ]) studied the asymptotic behavior, as ε tends to zero, of solutions of scalar boundary value problems defined in a domain ε obtained by removing from closed smoothed cubes well contained in (the holes) of diameter r(ε) ≤ ε periodically distributed with period ε in Rn. In particular in [ ] is studied, perhaps for the first time, a problem in which both Neumann and Dirichlet conditions are present on the boundary of the holes.
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