The point interaction approximation for diffusion in regions with many small holes

Let u(x,t)be the temperature at position x in R3 and at time t≥0. For N=1,2,3,... let w₁N,w₂N,...,w _N N, be a sequence of points in R3 and let DN be the domain in R3 defined by 1$${D^N} = \bigcap\limits_{j = 1}^N {\left( {x\left| {\left| {x - w_j^N} \right|} \right. > \frac{\alpha }{N}} \right)} $$ where α is a fixed positive constant. The temperature u satisfies the initial-boundary value problem 2$$\frac{{\partial u}}{{\partial t}} = \Delta uin{D^N},t > 0$$3$$u(x,0) = f(x)forxin{D^N}andu(x,t) = 0\;on\;B_j^Nfort > 0,j = 1,2,3,...,N$$ Here f (x) is a smooth, positive function of compact support representing the initial temperature distribution and B _j N is the sphere centered at w _j N with radius α/N.