Asymptotics of elliptic and parabolic PDEs. And their applications in statistical physics, computational neuroscience, and biophysics
The monograph under review deals with asymptotic methods for the construction of solutions to boundary value problems for partial differential equations arising in applications, as molecular and cellular biology and biophysics. Particular emphasis is on mixed boundary value problems and on the so-called narrow escape problem. The presentation of the methods is within the framework of Brownian motion and diffusion of particles. The book is organized in 12 chapters. Chapters 1--5 form the first part of the book, devoted to singular perturbations of elliptic boundary value problems, whereas the remaining chapters form the second (and last) part on mixed boundary conditions for elliptic and parabolic equations. Chapter 1 contains an introduction to second order elliptic boundary value problems containing a small leading part. In Chapter 2, the authors present elementary asymptotic approximations for the solutions to singularly perturbed linear ordinary differential equations, whereas in Chapter 3 they consider singular perturbation problems in elliptic and parabolic equations in higher dimension. In Chapter 4 they turn to analyze the eigenvalues of a non-self-adjoint elliptic operator. The first part is then completed by Chapter 5 devoted to short time asymptotics of the heat kernel. Chapter 6 opens the second part of the book with the mixed boundary value problem and the narrow escape time problem. In Chapter 7 they consider the mixed problem in \(\mathbb{R}^2\), while in Chapter 8 they study the narrow escape in \(\mathbb{R}^3\). Chapter 9 contains short-time asymptotics of the heat kernal and statistics for the narrow escape time. The aim of Chapter 10 is to extend the narrow escape time method to the construction of an asymptotic solution to the stationary Poisson-Nernst-Planck equations in a ball. Chapter 11 shows how the data can be used to reconstruct the surface diffusion. Chapter 12, the last one, is devoted to asymptotic formulas in molecular and cellular biology. Finally, a bibliography of 17 pages and a detailed analytical index conclude the volume. We also observe that almost each chapter contains a section of Annotations, where a discussion on the scientific literature related to the results is presented. The monograph is well written, interesting, and surely recommended to applied mathematicians, engineers, physicists, chemists, and neuroscientists interested into analytical methods for the asymptotic analysis of elliptic and parabolic PDEs of relevance in applications.
singular perturbations of elliptic boundary value problems
scientific article; zbMATH DE number 6855990