Asymptotic Analysis

We prove the existence of a solution to a problem modeling the stationary heat distribution in an inhomogeneous plane with a crack and derive explicit representations of singular terms of an asymptotic expansion of the heat flow in the vicinity of the crack tips.

We address the homogenization of the stationary diffusion equation in a composite medium with two components M ε and B ε having respectively A ε ( x ) and α ε A ε ( x ) as diffusivity coefficients. We assume periodic distribution with size ε of the “holes” B ε but no periodicity is assumed on the matrices A ε . The high contrast between the two components is characterized by the assumption that the sequence α ε decreases towards zero. We study the three regimes corresponding to the limits α : = lim ε → 0 α ε ε . It is shown in particular that in the case α = 0 , the inclusions B ε behave as holes on the macroscopic diffusion process.

The fast reaction limit of a volume–surface reaction–diffusion system is rigorously investigated. The system is motivated by proteins localisation in stem cell division. By using Ball’s energy equation method, we show that as the reaction rate constant goes to infinity, the solution of the original system converges to the solution of a heat equation with dynamical boundary condition. As a consequence, the dynamical boundary condition can be interpreted as a fast reaction limit of a volume–surface reaction–diffusion system.

We estimate the ground state energy for the magnetic Laplacian with a Robin boundary condition. In a special asymptotic limit, we find that the magnetic field does not contribute to the two-term expansion of the ground state energy, thereby proving that the Robin boundary condition weakens diamagnetism. We discuss a semi-classical version of the operator and prove that the ground states concentrate near the boundary points of maximal curvature.