##
**Parabolic quasilinear equations minimizing linear growth functionals.**
*(English)*
Zbl 1053.35002

Progress in Mathematics (Boston, Mass.) 223. Basel: Birkhäuser (ISBN 3-7643-6619-2/hbk). xiv, 340 p. (2004).

This book is devoted to PDE’s of elliptic and parabolic type associated to functionals having a linear growth in the gradient, with a special emphasis on the applications related to image restoration and nonlinear filters. The main model is the total variation functional (i.e. the \(L^1\) norm of the gradient, in a \(C^1\) setting) proposed in a seminal paper by L. Rudin, S. Osher and E. Fatemi in 1992 [cf. Physica D 60, No. 1–4, 259–268 (1992; Zbl 0780.49028)].

Chapter 1 contains a review of the variational approach to image restoration based on the total variation minimization, with a description of the Euler-Lagrange equation and of the constraints involved. It contains also some numerical computations and a review of the main numerical methods adopted in the literature.

Chapter 2 deals with the parabolic initial value problem \[ \begin{aligned} \frac{\partial u}{\partial t}= \text{div}\biggl( {Du\over | Du| }\biggr)&\quad \text{in }(0,\infty)\times\Omega,\\ u(0,x)=u_0(x)& \quad \text{in }\Omega\end{aligned} \] with Neumann boundary conditions, in a bounded domain \(\Omega\) of \({\mathbb R}^n\). The main result is the existence and the uniqueness of a weak solution for any initial datum in \(L^1\). Here “weak” refers mainly to the fact that \(D u/| D u| \) is not everywhere defined, so that even distributional solutions have to be properly defined. The proof is mainly based on the Crandall-Liggett generation theorem.

Chapter 3 deals, instead, with the same problem in \(\Omega={\mathbb R}^n\). After an initial discussion about the existence of strong solutions when the initial datum is in \(L^2({\mathbb R}^n)\), the main result is the existence, uniqueness and stability of entropy solutions (in the sense first introduced by Kruzhkov for scalar conservation laws) when the initial data are in \(L^1_{\text{loc}}({\mathbb R}^n)\). Also the case of measures for initial data is discussed.

Chapter 4 is devoted to the asymptotic properties of solutions of the problems discussed in the previous two chapters, and their qualitative properties. It is for instance shown that the flow decreases the surface area of level sets and that local maxima (resp. minima) strictly decrease (resp. increase) with time. Radial solutions are then computed, and for compactly supported initial conditions some extinction profiles are explicitly given.

Chapter 5 is devoted to the Dirichlet problem \[ \begin{aligned} \frac{\partial u}{\partial t}= \text{div}\biggl({Du\over | Du|}\biggr)&\quad \text{in }(0,\infty)\times\Omega,\\ u(t,x)=\varphi(x) &\quad \text{in }(0,\infty)\times\partial\Omega,\\ u(0,x)=u_0(x) &\quad \text{in }\Omega\end{aligned} \] in a bounded domain \(\Omega\) of \({\mathbb R}^n\). The proof still uses, as in Chapter 2, the abstract theory of semigroups, but in this case some \(L^1\) estimates are missing and the notion of entropy solution becomes crucial. The final result gives and existence and uniqueness for initial data in \(L^1(\Omega)\).

Chapter 6 and Chapter 7 contain the most recent and technical developments of this subject, concerned with PDE’s \[ \begin{aligned} \frac{\partial u}{\partial t}= \text{div\,}f_\xi(x,Du)&\quad \text{in }(0,\infty)\times\Omega,\\ u(0,x)=u_0(x) &\quad\text{in }\Omega\end{aligned} \] with Neumann boundary conditions, in a bounded domain \(\Omega\) of \({\mathbb R}^n\), arising from an energy functional \(\int_\Omega f(x,Du)\,dx\), with \(f\) having a linear growth in the gradient. Initial data in \(L^2(\Omega)\) are discussed in Chapter 6, obtaining existence and uniqueness of strong solutions. In Chapter 7 these results are extended to the \(L^1\) case, again relying on the concept of entropy solution.

The appendix of the book contains three review sections on nonlinear semigroups, functions of bounded variation and Anzellotti’s theory of normal traces for vector fields and their pairing with BV functions.

The book is written with great care, paying also a lot of attention to the bibliographical and historical notes. It is a recommended reading for all researchers interested in this field.

Chapter 1 contains a review of the variational approach to image restoration based on the total variation minimization, with a description of the Euler-Lagrange equation and of the constraints involved. It contains also some numerical computations and a review of the main numerical methods adopted in the literature.

Chapter 2 deals with the parabolic initial value problem \[ \begin{aligned} \frac{\partial u}{\partial t}= \text{div}\biggl( {Du\over | Du| }\biggr)&\quad \text{in }(0,\infty)\times\Omega,\\ u(0,x)=u_0(x)& \quad \text{in }\Omega\end{aligned} \] with Neumann boundary conditions, in a bounded domain \(\Omega\) of \({\mathbb R}^n\). The main result is the existence and the uniqueness of a weak solution for any initial datum in \(L^1\). Here “weak” refers mainly to the fact that \(D u/| D u| \) is not everywhere defined, so that even distributional solutions have to be properly defined. The proof is mainly based on the Crandall-Liggett generation theorem.

Chapter 3 deals, instead, with the same problem in \(\Omega={\mathbb R}^n\). After an initial discussion about the existence of strong solutions when the initial datum is in \(L^2({\mathbb R}^n)\), the main result is the existence, uniqueness and stability of entropy solutions (in the sense first introduced by Kruzhkov for scalar conservation laws) when the initial data are in \(L^1_{\text{loc}}({\mathbb R}^n)\). Also the case of measures for initial data is discussed.

Chapter 4 is devoted to the asymptotic properties of solutions of the problems discussed in the previous two chapters, and their qualitative properties. It is for instance shown that the flow decreases the surface area of level sets and that local maxima (resp. minima) strictly decrease (resp. increase) with time. Radial solutions are then computed, and for compactly supported initial conditions some extinction profiles are explicitly given.

Chapter 5 is devoted to the Dirichlet problem \[ \begin{aligned} \frac{\partial u}{\partial t}= \text{div}\biggl({Du\over | Du|}\biggr)&\quad \text{in }(0,\infty)\times\Omega,\\ u(t,x)=\varphi(x) &\quad \text{in }(0,\infty)\times\partial\Omega,\\ u(0,x)=u_0(x) &\quad \text{in }\Omega\end{aligned} \] in a bounded domain \(\Omega\) of \({\mathbb R}^n\). The proof still uses, as in Chapter 2, the abstract theory of semigroups, but in this case some \(L^1\) estimates are missing and the notion of entropy solution becomes crucial. The final result gives and existence and uniqueness for initial data in \(L^1(\Omega)\).

Chapter 6 and Chapter 7 contain the most recent and technical developments of this subject, concerned with PDE’s \[ \begin{aligned} \frac{\partial u}{\partial t}= \text{div\,}f_\xi(x,Du)&\quad \text{in }(0,\infty)\times\Omega,\\ u(0,x)=u_0(x) &\quad\text{in }\Omega\end{aligned} \] with Neumann boundary conditions, in a bounded domain \(\Omega\) of \({\mathbb R}^n\), arising from an energy functional \(\int_\Omega f(x,Du)\,dx\), with \(f\) having a linear growth in the gradient. Initial data in \(L^2(\Omega)\) are discussed in Chapter 6, obtaining existence and uniqueness of strong solutions. In Chapter 7 these results are extended to the \(L^1\) case, again relying on the concept of entropy solution.

The appendix of the book contains three review sections on nonlinear semigroups, functions of bounded variation and Anzellotti’s theory of normal traces for vector fields and their pairing with BV functions.

The book is written with great care, paying also a lot of attention to the bibliographical and historical notes. It is a recommended reading for all researchers interested in this field.

Reviewer: Luigi Ambrosio (Pisa)

### MSC:

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

35K55 | Nonlinear parabolic equations |

49J10 | Existence theories for free problems in two or more independent variables |

49K10 | Optimality conditions for free problems in two or more independent variables |

68U10 | Computing methodologies for image processing |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |

47H20 | Semigroups of nonlinear operators |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |