Homogenization of a Continuously Microperiodic Multidimensional Medium

The asymptotic homogenization method is applied to obtain formal asymptotic solution and the homogenized solution of a Dirichlet boundary-value problem for an elliptic equation with rapidly os- cillating coefficients. The proximity of the formal asymptotic solution and the homogenized solution to the exact solution is proved, which provides the mathematical justification of the homogenization pro- cess. Preservation of the symmetry and positive-definiteness of the effective coefficient in the homogenized problem is also proved. An example is presented in order to illustrate the theoretical results.

Singularly Perturbed Systems with Diffusion and Time Delay

The singularly perturbed systems for the nonlinear three-species food-chain reaction systems with time delay are considered. By using the method of the stretched variable, the formal asymptotic solution is obtained under suitable conditions.

Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition

Minimizers and gradient flows are studied for the functional ∫ Ω W(u) + ϵ 2 ∣∇ u ∣ 2 d x , Ω ⊆ R n , ϵ > 0, where u satisfies a Dirichlet condition u = h ϵ on ∂ Ω . Here W is taken to be a double-well potential with minimum value zero attained at u = a and u = b . Questions of existence and structure of minimizers for small ϵ are resolved through the identification of a limiting variational problem, the so-called Γ-limit. A formal asymptotic solution is then constructed for the gradient flow ∂ t u ϵ = 2 ϵ ∆ u ϵ — ϵ -1 W' ( u ϵ ), u ϵ ( x , 0) = g ( x ), u ϵ ( x, t ) = h ϵ on ∂ Ω , valid when ϵ is small. Using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity ϵk , where k is mean curvature. At the intersection of a front with ∂ Ω , the Dirichlet condition is shown to imply a contact angle condition for the front. This asymptotically correct evolution process represents gradient flow for the Γ-limit.