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Partial differential equations. A unified Hilbert space approach. (English) Zbl 1275.35002

de Gruyter Expositions in Mathematics 55. Berlin: de Gruyter (ISBN 978-3-11-025026-8/hbk; 978-3-11-025027-5/ebook). xviii, 469 p. (2011).
The authors study partial differential equations from the standpoint of Sobolev chains and Sobolev lattices. Let \(C\subseteq H\oplus H\) be a densely defined, closed linear operator in a Hilbert space \(H\) with \(0\) in the resolvent set \(\rho(C)\). Then the triple \((H_1(C), H,H_{-1}(C))\), denoting \(H_1(C)= D(C)\) with norm \(|C\cdot|_H\), and \(H_{-1}(C)= D(C^*)\) with norm \(|C^*\cdot|_H\), is called a short Sobolev chain. \(D(C^n)\) equipped with the norm \(|C^n\cdot|_H\), is denoted by \(H_n(C)\). \(H_{-n}(C):=H_n(C^*)^*\). The family \((H_n(C))_{n\in\mathbb{Z}}\) is called the long Sobolev chain associated with \(C\). \(H_{-\infty}(C)= \bigcup_{k\in\mathbb{Z}}H_k(C)\), and \(H_\infty(C)=\bigcap_{k\in\mathbb{Z}} H_k(C)\).
Let \(D= 2^{-1/2}(m-\partial)\), \(D^*= 2^{-1/2}(m+\partial)\). Starting from the base space \(L_2(\mathbb{R},\) \(\exp(-2\nu x) dx)\), the authors define \(D_\nu:= \exp(\nu m)D\exp(-\nu m)\), \(D^*_\nu:\exp(\nu m)D^*\exp(-\nu m)\), and give the Sobolev chain associated with \(|D_\nu|\), \(\nu\in\mathbb{R}\), in \(\mathbb{R}\). They also define the Sobolev lattice for the family \((H_\alpha(C); \alpha\in\mathbb{Z}^{n+1})\) with \(C= (C_i)_{i=0,\dots, n}\), and obtain the relation \((\partial_\nu\pm 1)^k= L^*_\nu(im\pm 1)^k L_\nu\), by using Fourier-Laplace transform \(L_\nu\). They treat linear PDEs with constant coefficients in \(H_{-\infty}(\partial_\nu+ e)\) and in \(H_{-\infty}(|D_\nu|)\) in Chapter 3. Although evolution equations of mathematical physics (MP) are properly treated in Chapter 5, we find many examples appearing in MP, and fundamental solutions of them, in Chapter 3. They treat “Causality” in 3.1.10 and 4.2.2. That is, let \(\nu\) be a direction in \(\mathbb{R}^{n+1}\). For a linear functional \(f\) on \(C°_\infty(\mathbb R^{n+1})\) define \[ \text{supp}_{\nu} f:=\mathbb{R}\setminus\bigcup \{I\mid(I\text{ open in }\mathbb{R})\wedge (f= 0\text{ in }[I]_{\nu}+ \{\nu\}^\perp)\}. \] Let \(W\) be a mapping from a subset of linear functionals on \(C°_\infty(\mathbb{R}^{n+1})\) with values in such subset. \(W\) is causal in direction \(\nu\) if \[ \inf\text{supp}_{\nu(0)}(f- g)\leq \inf\text{supp}_{\nu(0)}(W(f)- W(g)). \]
In Chapter 6, they treat evolutionary problems with material laws. Let \(A\) be a skew self-adjoint operator. They aim to find \(U,V\in H_{-\infty}(\partial_\nu+ \nu)\otimes H\) such that, for a given \(f\in H_{-\infty}(\partial_\nu+\nu)\otimes H\), \(\partial_0 V+ AU= f\) and \(V= M(\partial^{-1}_0)U\) hold. \(M(\partial^{-1}_0):= L^*_\nu M(1/(im_0+ \nu)) L_\nu,\) \(\nu> 1/(2r)\), is forward causal, if \(\{M(z);\;z\in B_{C(r,r)}\}\) is a holomorphic family of uniformly bounded linear operators on \(H\). \(B_{C(r,r)}\to [i\mathbb{R}]+ [R_{>1/(2r)}]\), \(z\to 1/z\), is a bijection for \(r\in\mathbb{R}_{> 0}\).

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
80A20 Heat and mass transfer, heat flow (MSC2010)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
47D06 One-parameter semigroups and linear evolution equations
47A60 Functional calculus for linear operators
35Qxx Partial differential equations of mathematical physics and other areas of application
35A08 Fundamental solutions to PDEs
35G05 Linear higher-order PDEs
35C15 Integral representations of solutions to PDEs
35G35 Systems of linear higher-order PDEs
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