##
**Differential equations with operator coefficients with applications to boundary value problems for partial differential equations.**
*(English)*
Zbl 0920.35003

Springer Monographs in Mathematics. Berlin: Springer. xx, 442 p. (1999).

When solving partial differential equations, one of the most successful classical methods is to reduce the number of variables by looking for solutions of a particular shape in one variable, say \(t\), namely \(y(x,t)=v(x)e^{i\lambda t}\), where \(x\) stands for the remaining variables. Substitution of \(y(x,t)\) into the given partial differential equation leads to a (partial) differential equation in the remaining variables \(x\) depending polynomially on an eigenvalue parameter \(\lambda\). This can be considered as the starting point of spectral theory. In order to solve spectral problems properly it is necessary to define a suitable operator. It is advantageous to have a representation by bounded operators. This yields a spectral problem for an operator polynomial with bounded differential operators, defined in suitable Sobolev spaces. But for considering this type of spectral problem it is not necessary to have differential operators; bounded operators will do.

This is the starting point of the monograph under review. The differential equation with operator coefficients \[ L(t,D_t)u(t):=\sum_{q=0}^\ell A_{\ell-q}(t)D_t^q u(t)=f(t) \] is considered, where the operators \(A_j(t)\) are bounded from a Hilbert space \(\mathcal H\) to a Hilbert space \(H\). First the case that the operators \(A_j\) are independent of \(t\) is investigated and it is assumed that the operator polynomial \[ {\mathcal A}(\lambda)=\sum_{q=0}^\ell A_{\ell-q}\lambda^q \] is Fredholm and that \({\mathcal A}(\lambda)\) is invertible for at least one \(\lambda\). Then a solution \(W(t)\) of \[ {\mathcal A}(D_t)W(t)=e^{i\lambda_0t}\sum_{0\leq j\leq N}\frac{t^j}{j!} y_{N-j}\quad \text{on} \mathbb R \] exists and is of the form \[ W(t)=e^{i\lambda_0t}\sum_{0\leq j\leq N+m}\frac{t^j}{j!}x_{N+m-j}, \] where \(x_j\in {\mathcal H}\), \(j=0,\dots,N+m\), \(m=0\) if \({\mathcal A}(\lambda_0)\) is invertible and \(m\) the maximal length of a Jordan chain of \({\mathcal A}\) at \(\lambda_0\) otherwise. Chapter 1 ends with various applications to elliptic partial differential equations. Introducing weighted Lebesgue and Sobolev spaces \(L_{2,\beta}(\mathbb R;H_0)\) and \(W_\beta^\ell(\mathbb R)\) with weight \(e^{\beta t}\) and values in suitable Banach spaces it is shown that \[ {\mathcal A}(D_t):W_\beta^\ell(\mathbb R)\to L_{2,\beta}(\mathbb R;H_0) \] is invertible if \({\mathcal A}(\lambda)\) has no eigenvalues on the line \(\operatorname{Im} \lambda=\beta\). In that case the solution \(u\) of \({\mathcal A}(D_t)u=f\) can be written as \[ u(t)=\frac 1{2\pi}\int_{\operatorname{Im} \lambda=\beta}e^{it\lambda}{\mathcal A}^{-1} (\lambda)\tilde f(\lambda) d\lambda, \] where \(\tilde f(\lambda)=\int_{\mathbb R}e^{-i\lambda t}f(t) dt\) is the Fourier transform of \(f\). This gives rise to the definition of the Green’s kernel \[ G(t)=\frac 1{2\pi}\int_{\operatorname{Im} \lambda=\beta}e^{it\lambda}{\mathcal A}^{-1} (\lambda) d\lambda. \] An asymptotic representation of \(G(t-\tau)\) leads to asymptotic representations of solutions (Section 2.8.1). This is complemented by estimates of solutions on finite intervals (Section 2.9).

In Chapter 3 solutions of \({\mathcal A}(D_t)u=0\) are investigated, including the questions of uniqueness of the solutions of the homogeneous equation. It is worth-while to note that the ordinary differential equation \[ (-1)^{m_-}(\partial_t+k_+)^{m_+}(\partial_t+k_-)^{m_-}w(t) =\| f\| _{L_2(t,t+1;H_0)} \] is considered together with \({\mathcal A}(D_t)u=f\), where the numbers \(k_-\), \(k_+\) are such that the strip \(k_-<\operatorname{Im} \lambda<k_+\) does not contain eigenvalues of \({\mathcal A}(\lambda)\) and where \(m_\pm\) is the maximum of the lengths of the Jordan chains corresponding to the eigenvalues on the line \(\operatorname{Im} \lambda=k_{\pm}\) (\(m_\pm\) being \(1\) if there are no eigenvalues). This comparison differential equations plays a crucial role throughout the monograph. In Chapter 4 solvability in spaces with a more general weight is considered. In Chapters 5 to 10 these results are generalized to differential equations with variable operator coefficients. Chapters 11 to 17 are concerned with the asymptotics of solutions of perturbations of \({\mathcal A}(D_t)\). Some particular perturbations are considered in Chapter 11. As an example, in Section 11.6 an elliptic operator with constant coefficients on a cuspidal domain is transformed into an operator with variable coefficients on a semicylinder, and for the Dirichlet problem it is shown that solutions of the homogeneous equation satisfy certain explicitly given asymptotics. In Chapter 12 the unperturbed pencil \({\mathcal A}(\lambda)\) is reduced to a \(\lambda\)-linear pencil in the usual way, which corresponds to the reduction of \({\mathcal A}(D_t)\) to a first-order system. Splitting off a finite-dimensional part of the first-order system by using the Riesz projection for the finitely many eigenvalues on a line \(\operatorname{Im} \lambda=k_0\), solutions for both parts are investigated separately (Chapter 13). Finally, Chapters 14 to 17 deal in detail with power-exponential asymptotics and cases of particular eigenvalue behaviour on the line \(\operatorname{Im} \lambda=k_0\), respectively.

In summary, this well-written monograph contains a wealth of results on differential equations with operator coefficients. Despite the technical nature of many of the results, the exposition is well structured and clear. Many examples emphasize the importance of the presented results and will help the interested researcher to apply the abstract results to many more problems.

This is the starting point of the monograph under review. The differential equation with operator coefficients \[ L(t,D_t)u(t):=\sum_{q=0}^\ell A_{\ell-q}(t)D_t^q u(t)=f(t) \] is considered, where the operators \(A_j(t)\) are bounded from a Hilbert space \(\mathcal H\) to a Hilbert space \(H\). First the case that the operators \(A_j\) are independent of \(t\) is investigated and it is assumed that the operator polynomial \[ {\mathcal A}(\lambda)=\sum_{q=0}^\ell A_{\ell-q}\lambda^q \] is Fredholm and that \({\mathcal A}(\lambda)\) is invertible for at least one \(\lambda\). Then a solution \(W(t)\) of \[ {\mathcal A}(D_t)W(t)=e^{i\lambda_0t}\sum_{0\leq j\leq N}\frac{t^j}{j!} y_{N-j}\quad \text{on} \mathbb R \] exists and is of the form \[ W(t)=e^{i\lambda_0t}\sum_{0\leq j\leq N+m}\frac{t^j}{j!}x_{N+m-j}, \] where \(x_j\in {\mathcal H}\), \(j=0,\dots,N+m\), \(m=0\) if \({\mathcal A}(\lambda_0)\) is invertible and \(m\) the maximal length of a Jordan chain of \({\mathcal A}\) at \(\lambda_0\) otherwise. Chapter 1 ends with various applications to elliptic partial differential equations. Introducing weighted Lebesgue and Sobolev spaces \(L_{2,\beta}(\mathbb R;H_0)\) and \(W_\beta^\ell(\mathbb R)\) with weight \(e^{\beta t}\) and values in suitable Banach spaces it is shown that \[ {\mathcal A}(D_t):W_\beta^\ell(\mathbb R)\to L_{2,\beta}(\mathbb R;H_0) \] is invertible if \({\mathcal A}(\lambda)\) has no eigenvalues on the line \(\operatorname{Im} \lambda=\beta\). In that case the solution \(u\) of \({\mathcal A}(D_t)u=f\) can be written as \[ u(t)=\frac 1{2\pi}\int_{\operatorname{Im} \lambda=\beta}e^{it\lambda}{\mathcal A}^{-1} (\lambda)\tilde f(\lambda) d\lambda, \] where \(\tilde f(\lambda)=\int_{\mathbb R}e^{-i\lambda t}f(t) dt\) is the Fourier transform of \(f\). This gives rise to the definition of the Green’s kernel \[ G(t)=\frac 1{2\pi}\int_{\operatorname{Im} \lambda=\beta}e^{it\lambda}{\mathcal A}^{-1} (\lambda) d\lambda. \] An asymptotic representation of \(G(t-\tau)\) leads to asymptotic representations of solutions (Section 2.8.1). This is complemented by estimates of solutions on finite intervals (Section 2.9).

In Chapter 3 solutions of \({\mathcal A}(D_t)u=0\) are investigated, including the questions of uniqueness of the solutions of the homogeneous equation. It is worth-while to note that the ordinary differential equation \[ (-1)^{m_-}(\partial_t+k_+)^{m_+}(\partial_t+k_-)^{m_-}w(t) =\| f\| _{L_2(t,t+1;H_0)} \] is considered together with \({\mathcal A}(D_t)u=f\), where the numbers \(k_-\), \(k_+\) are such that the strip \(k_-<\operatorname{Im} \lambda<k_+\) does not contain eigenvalues of \({\mathcal A}(\lambda)\) and where \(m_\pm\) is the maximum of the lengths of the Jordan chains corresponding to the eigenvalues on the line \(\operatorname{Im} \lambda=k_{\pm}\) (\(m_\pm\) being \(1\) if there are no eigenvalues). This comparison differential equations plays a crucial role throughout the monograph. In Chapter 4 solvability in spaces with a more general weight is considered. In Chapters 5 to 10 these results are generalized to differential equations with variable operator coefficients. Chapters 11 to 17 are concerned with the asymptotics of solutions of perturbations of \({\mathcal A}(D_t)\). Some particular perturbations are considered in Chapter 11. As an example, in Section 11.6 an elliptic operator with constant coefficients on a cuspidal domain is transformed into an operator with variable coefficients on a semicylinder, and for the Dirichlet problem it is shown that solutions of the homogeneous equation satisfy certain explicitly given asymptotics. In Chapter 12 the unperturbed pencil \({\mathcal A}(\lambda)\) is reduced to a \(\lambda\)-linear pencil in the usual way, which corresponds to the reduction of \({\mathcal A}(D_t)\) to a first-order system. Splitting off a finite-dimensional part of the first-order system by using the Riesz projection for the finitely many eigenvalues on a line \(\operatorname{Im} \lambda=k_0\), solutions for both parts are investigated separately (Chapter 13). Finally, Chapters 14 to 17 deal in detail with power-exponential asymptotics and cases of particular eigenvalue behaviour on the line \(\operatorname{Im} \lambda=k_0\), respectively.

In summary, this well-written monograph contains a wealth of results on differential equations with operator coefficients. Despite the technical nature of many of the results, the exposition is well structured and clear. Many examples emphasize the importance of the presented results and will help the interested researcher to apply the abstract results to many more problems.

Reviewer: Manfred MĂ¶ller (Johannesburg)