THE GRADIENT THEORY OF THE PHASE-TRANSITIONS IN CAHN-HILLIARD FLUIDS WITH DIRICHLET BOUNDARY-CONDITIONS

We are interested in the asymptotic behavior of minimizers (as epsilon --> 0) of the variational problems under the Dirichlet condition inf{ integral(Omega)[epsilon\del u\(2) + 1/epsilon W(x,u)] dx \ u is an ele ment of W-1,W-2(Omega : R(n)), u = g on partial derivative Omega}, whe re W(x,.) is a nonnegative function with only two zeros alpha and beta . Here alpha and beta are independent of the space variable x. In this paper, we will show that the limit of a sequence of minimizers {u(eps ilon)}(epsilon>0) (as epsilon --> 0) is a solution of another variatio nal problem without boundary condition. However the limit variational problem has a boundary integral corresponding to transition layers nea r partial derivative Omega. Our analysis relies mainly on the theory o f gamma convergence. In order to overcome the difficulty of inhomogene ity of the boundary condition, we approximate g(x) by suitable piecewi se smooth functions near the boundary partial derivative Omega.