Second order partial differential equations in Hilbert spaces.

*(English)*Zbl 1012.35001
London Mathematical Society Lecture Note Series. 293. Cambridge: Cambridge University Press. xvi, 348 p. (2002).

The book has as a main object the linear elliptic and parabolic equations of second order on an infinite-dimensional separable Hilbert space \(H\) such as
\[
\lambda\psi(x) - \tfrac 12 \text{Tr}[Q(x) D^2\psi(x)] - \langle F(x), D\psi(x)\rangle = g(x),\quad x\in H,
\]
and
\[
D_t u(t,x) = \tfrac 12 \text{Tr}[Q(x) D^2\psi(x)] + \langle F(x), D\psi(x)\rangle,\quad x\in H,\quad t>0,
\]

\[ u(0,x) = \phi(x),\quad x\in H, \] where \(F:D(F)\rightarrow H\), \(Q:D(Q)\rightarrow H\) and \(\phi,g:H\rightarrow \mathbb{R}\) are given mappings.

In part I the authors deal with the case where \(F\) and \(Q\) are continuous and work on the space \(UC_b(H)\) of all uniformly continuous bounded functions from \(H\) into \(\mathbb{R}\). Chapters 1 and 2 are devoted to Gaussian measure and continuous functions on \(H\). Chapter 3 is devoted to the problem \[ D_t u(t,x) = \tfrac 12\text{ Tr}[Q(x) D^2\psi(x)],\;\;x\in H,\;t>0, \]

\[ u(0,x) = \phi(x),\;\;x\in H. \] The authors prove that the assumption that \(Q\) is of trace class is necessary to have a solution of the problem at least for initial data \(\phi\) smooth enough. In this case they study existence, uniqueness and regularity of the solution in \(UC_b(H)\) to the above problem; in particular they show that the solutions are smooth in the directions of the reproducing kernel \(Q^{{1}\over{2}}(H)\). They also investigate the infinitesimal generator of the corresponding semigroup. In Chapter 4 the elliptic problem \[ \lambda\psi(x) - \tfrac 12 \text{Tr}[Q(x) D^2\psi(x)] - \langle F(x), D\psi(x)\rangle = g(x),\quad x\in H, \] is investigated: existence and uniqueness of a solution and also Schauder type estimates are proved. In Chapter 5 the authors consider the case where \(F\) and \(Q\) are bounded and Hölder continuous. Here the results are not so satisfactory as in the finite-dimensional case. In particular if \(g\) is Hölder continuous, then first and second derivatives are Hölder continuous, but nothing is known on the trace of \(QD^2\psi\). In Chapter 6 the Ornstein-Uhlenbeck operator is considered as the typical example of the case in which \(f\) and \(Q\) are unbounded. The transition semigroup is studied giving conditions to have a strong Feller property. The infinitesimal generator of the semigroup is studied as well as several properties including Schauder estimates. Chapter 7 is devoted to a general Kolmogorov equation (under rather strong conditions on the data). Existence and uniqueness results are proved by the method of stochastic characteristics, and regularity results are presented in Section 7.5 and 7.6.

In part II the authors consider the case of very irregular unbounded coefficients, where it is useful to work in the spaces \(L^2(H,\nu)\) where \(\nu\) is an invariant Gaussian measure. In Chapter 10 the case of Ornstein-Uhlenbeck semigroup is considered showing hypercontractivity, Poincaré and log-Sobolev inequalities with some of their consequences. Chapters 11 and 12 are devoted to perturbations of Ornstein-Uhlenbeck generator of the type \(\langle F(x), D\psi(x)\rangle\) with \(F\) bounded or Lipschitz continuous (Chapter 11) or the more general case of the form \(F(x) = - DU(x)\) (Chapter 12).

Part III is devoted to applications in control theory giving Hamilton-Jacobi type equations and inequalities.

\[ u(0,x) = \phi(x),\quad x\in H, \] where \(F:D(F)\rightarrow H\), \(Q:D(Q)\rightarrow H\) and \(\phi,g:H\rightarrow \mathbb{R}\) are given mappings.

In part I the authors deal with the case where \(F\) and \(Q\) are continuous and work on the space \(UC_b(H)\) of all uniformly continuous bounded functions from \(H\) into \(\mathbb{R}\). Chapters 1 and 2 are devoted to Gaussian measure and continuous functions on \(H\). Chapter 3 is devoted to the problem \[ D_t u(t,x) = \tfrac 12\text{ Tr}[Q(x) D^2\psi(x)],\;\;x\in H,\;t>0, \]

\[ u(0,x) = \phi(x),\;\;x\in H. \] The authors prove that the assumption that \(Q\) is of trace class is necessary to have a solution of the problem at least for initial data \(\phi\) smooth enough. In this case they study existence, uniqueness and regularity of the solution in \(UC_b(H)\) to the above problem; in particular they show that the solutions are smooth in the directions of the reproducing kernel \(Q^{{1}\over{2}}(H)\). They also investigate the infinitesimal generator of the corresponding semigroup. In Chapter 4 the elliptic problem \[ \lambda\psi(x) - \tfrac 12 \text{Tr}[Q(x) D^2\psi(x)] - \langle F(x), D\psi(x)\rangle = g(x),\quad x\in H, \] is investigated: existence and uniqueness of a solution and also Schauder type estimates are proved. In Chapter 5 the authors consider the case where \(F\) and \(Q\) are bounded and Hölder continuous. Here the results are not so satisfactory as in the finite-dimensional case. In particular if \(g\) is Hölder continuous, then first and second derivatives are Hölder continuous, but nothing is known on the trace of \(QD^2\psi\). In Chapter 6 the Ornstein-Uhlenbeck operator is considered as the typical example of the case in which \(f\) and \(Q\) are unbounded. The transition semigroup is studied giving conditions to have a strong Feller property. The infinitesimal generator of the semigroup is studied as well as several properties including Schauder estimates. Chapter 7 is devoted to a general Kolmogorov equation (under rather strong conditions on the data). Existence and uniqueness results are proved by the method of stochastic characteristics, and regularity results are presented in Section 7.5 and 7.6.

In part II the authors consider the case of very irregular unbounded coefficients, where it is useful to work in the spaces \(L^2(H,\nu)\) where \(\nu\) is an invariant Gaussian measure. In Chapter 10 the case of Ornstein-Uhlenbeck semigroup is considered showing hypercontractivity, Poincaré and log-Sobolev inequalities with some of their consequences. Chapters 11 and 12 are devoted to perturbations of Ornstein-Uhlenbeck generator of the type \(\langle F(x), D\psi(x)\rangle\) with \(F\) bounded or Lipschitz continuous (Chapter 11) or the more general case of the form \(F(x) = - DU(x)\) (Chapter 12).

Part III is devoted to applications in control theory giving Hamilton-Jacobi type equations and inequalities.

Reviewer: Marco Biroli (Monza)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |