Functional calculus of pseudo-differential boundary problems
Let \(\Omega\) be an open subset of \({\mathbb{R}}^ n\) with smooth boundary \(\partial \Omega\). Let us consider the boundary value problem \[ (1)\quad \Delta u=f\quad in\quad \Omega,\quad Tu=\phi \quad at\quad \partial \Omega, \] where \(\Delta\) is the Laplacian and T a trace operator (e.g. the usual trace on \(\partial \Omega)\). If R, K are the operators solving (1) with \(\phi =0\), resp. \(f=0\), then the operator \(\left( \begin{matrix} \Delta \\ T\end{matrix} \right)\) is inverted by (R,K). Actually R is equal to \(Q_{\Omega}+G\), where \(Q_{\Omega}\) is a restriction to \(\Omega\) of an inverse Q of \(\Delta\), and G is a special term adapted to the boundary condition and called a singular Green operator. The consideration of Problem (1) above therefore motivates the study of general operators of the form \[ (2)\quad \left( \begin{matrix} P_{\Omega}+G\\ T\end{matrix} \begin{matrix} K\\ S\end{matrix} \right): C({\bar \Omega})^ N\times C^{\infty}(\partial \Omega)^ M\quad \to \quad C^{\infty}({\bar \Omega})^{N'}\times C^{\infty}(\partial \Omega)^{M'}\quad, \] where P and S are pseudo-differential operators, G is a singular Green operator and K is a so-called Poisson operator. Operators of type (2) were introduced and studied by Boutet de Monvel. The main purpose of the book under review is to set up an operational calculus for operators of type (2) defined from differential and pseudo- differential boundary value problems via a resolvent construction, and to present some applications of this to evolution problems, fractional powers, spectral theory and singular perturbation problems. Here is a quick (and superficial) description of the contents of this book. In Chapter 1, the theory of boundary problems of type (2) is surveyed, with special attention devoted to the case of the so-called normal boundary conditions. Operators of type (2) depending on a parameter is introduced. For them a symbolic calculus is developed in Chapter 2, and a parametrix and resolvent construction is given in Chapter 3. (This is not an easy consequence of the parameter independent theory). In Chapter 4, applications are discussed. First the solvability of parabolic problems of the form \[ \partial_ tu+P_{\Omega}u+Gu=f\quad in\quad \Omega,\quad t>0,\quad Tu=\phi \quad at\quad \partial \Omega,\quad t>0,\quad u|_{t=0}=u_ 0\quad in\quad \Omega, \] is discussed on the basis of the resolvent studied before in Chapter 3. If B is the operator \(P+G\) (with domain defined by the boundary condition \(Tu=0)\), a study is made of the heat operator exp(-tB); in particular a trace formula is given, with applications to index formulas. Complex powers \(B^ z\) and spectral theory for B are studied. Counting formulas for the eigenvalues of \((B^*B)^{1/2}\) are established. Applications to more general eigenvalue problems (so-called implicit problems of the form \(\lambda A_ 1u=A_ 0u\), with boundary conditions) are studied. Also applications to singular perturbation problems are presented. The subject is of course very technical; however the book is written in a very clear way. Many examples are treated, and comparisons are made between the author's results and work by other authors. This book is very useful for anyone interested in the theory of boundary problems for pseudo-differential operators and its applications.
scientific article; zbMATH DE number 4008910