Green's kernels and meso-scale approximations in perforated domains / Vladimir Maz'ya, Alexander Movchan, Michael Nieves

The book deals with the analysis of Green's functions of elliptic boundary value problems. The authors concentrate on the following two topics: (1) asymptotics of Green's functions and tensors for the Laplace and Lamé operators, respectively, in singularly perturbed domains, and (2) meso-scale asymptotics in non-periodically perforated domains.
The book consists of three parts. Part I is devoted to uniform asymptotics of Green's functions for the Laplace operator with a small hole or inclusion. Various combinations of boundary conditions on the boundary of hole and exterior boundary are considered. Asymptotic expansions for Green's functions in domains with several small inclusions are studied as well. The main ingredient of the authors' approach is the capacity potential.
In Part II, the authors study the uniform asymptotics of Green's tensor for a linear elasticity system in domains with one or a few small rigid inclusions or void holes. Here, the matrix of elastic capacity plays a crucial role.
Part III deals with the case of perforated domains that contain many inclusions or holes of different size. In contrast to the homogenization theory, no periodicity assumption is imposed. The authors introduce a novel method of meso-scale asymptotic approximation of solutions. An important part of their asymptotic expansion algorithm consists of boundary layers near individual holes. In the case of spherical holes all these boundary layer terms can be found explicitly. (MSN)