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Diffusion equations. Transl. from the Jap. by Seizô Itô. (English) Zbl 0787.35001
Translations of Mathematical Monographs. 114. Providence, RI: American Mathematical Society (AMS). x, 225 p. (1992).
This book by Prof. Itô, translated from the Japanese by himself, is very well represented by the author’s abstract: “In this book, the author presents diffusion equations with variable coefficients associated with boundary conditions and the corresponding elliptic boundary value problems. The fundamental solution of the initial boundary value problem and Green function of the elliptic boundary value problem are constructed, and the formulae that express solutions of those problems by using the fundamental solution or Green function are presented. Several important properties of the solutions are also discussed.”
The monograph is devoted to linear diffusion problems, and the differential operators have principal part in divergence form that includes the volume element of a Riemannian metric associated to this principal part (cf. Ch. 2,3). This device enables the author to treat boundary conditions employing local coordinate systems. All along the book the author derives as many propositions as possible using the fundamental solution of the diffusion equation: in fact, he proves hypoellipticity for elliptic and parabolic operators by employing the fundamental solution (cf. Ch. 4, §24). The contents of the book are as follows:
Introduction. Includes a short account on physical background of the diffusion process, and derivation of equations and boundary conditions. There is also a description of the results and notation.
Chapter 1. Fundamental Solutions of Diffusion Equations in Euclidean Spaces.
Here the fundamental solution of a general linear parabolic differential operator \[ a^{ij}(t,x) {\partial^ 2 \over \partial x^ i \partial x^ j} u+b^ i(t,x) {\partial \over \partial x^ i} u+c(t,x)u- {\partial \over \partial t} u \] and its adjoint, is investigated by using a parametrix method.
Chapter 2. Diffusion Equations in a Bounded Domain.
Here the operator is \[ {1 \over \sqrt{a(x)}}{\partial \over \partial x^ i}[\sqrt{a(x)}a^{ij}(x) {\partial u \over \partial x^ j}]+b^ i(t,x){\partial u \over \partial x^ i}+c(t,x)u-{\partial u\over\partial t},\quad a(x)= \text{det} | a^{ij}(x)|^{-1}, \] and the boundary condition is \(\alpha u-(1-\alpha) \partial u/\partial n=0,\alpha=\alpha(t,x)\).
For the extreme cases \(\alpha \equiv 1\), \(\alpha \equiv 0\), one has for the heat equation in a half space, the well known kernels \[ G(t,x;s,y)- G(t,x;s,\overline y),\;G(t,x;s,y)+G(t,x;s,\overline y),\;\overline y=(- y^ 1,y^ 2, \cdots,y^ n), \] for Dirichlet and Neumann problems; now the spatial domain is a bounded \(\Omega\) with \(C^ 3\) boundary and a construction of local coordinates gives the analogue of \(\overline y\) for general \(\alpha\). The chapter includes construction of fundamental solution for the operator and its adjoint via a partition of unity, its uniqueness and positiveness, existence and uniqueness of solution to initial-boundary value problems for the operator and its adjoint, maximum principles, and dependence of the solution on coefficients, boundary conditions and domains.
Chapter 3. Diffusion Equations in Unbounded Domains.
Construction of a fundamental solution in unbounded regions by approximation with a monotone sequence of fundamental solutions of expanding, bounded subregions. Nonuniqueness. Existence for nonhomogeneous initial-boundary value problems. This chapter includes also a short account on evolution operators in \(L^ p\) and \(C\) spaces, and eigenfunction expansions for the elliptic operators in bounded domains. The latter are used to give expansions of the fundamental solutions.
Chapter 4. Elliptic Boundary Value Problems.
Here the Green’s function for elliptic boundary value problems is obtained in general, not necessarily bounded domains, by employing the fundamental solutions of the associated parabolic operators.
Chapter 5. Some Related Topics in Vector Analysis.
Includes solenoidal and potential components of a vector field, and Helmholtz decomposition.
Supplementary notes and References are included, quoting 27 items.

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
35A08 Fundamental solutions to PDEs
35K10 Second-order parabolic equations
35C10 Series solutions to PDEs
35J15 Second-order elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B50 Maximum principles in context of PDEs
35C15 Integral representations of solutions to PDEs