##
**Generalized solutions of first-order PDEs. The dynamical optimization perspective.**
*(English)*
Zbl 0820.35003

Basel: Birkhäuser,. xi, 312 p. (1994).

In the author’s own words, “this book is intended as a self-contained exposition of the theory of minimax solutions” for some classes of first-order PDE of the form:
\[
F(x,u (x),Du(x)) = 0, \;\;x \in G = \text{Int} (G) \subset \mathbb{R}^ n, \quad u(x) = u_ 0(x)\;\forall x \in G_ 0 \subset \mathbb{R}^ n.\tag{1}
\]
The functions \(f(x,.,p): \mathbb{R} \to \mathbb{R}\), \((x,p) \in G \times \mathbb{R}^ n\), are assumed to be nonincreasing and in most of the results, for any \((x,z) \in G \times \mathbb{R}\), the function \(F(x,z,.)\) is assumed to be Lipschitzian of rank \(\rho (x,z)\) where \(\rho (.,.)\) is continuous.

The first definition of the author’s “minimax solution”, as a continuous function, \(u(.) : G \to \mathbb{R}\), such that for any \((x_ 0,z_ 0) \in \text{graph} u(.)\) and any \(p \in \mathbb{R}^ n\) there exist \(r>0\) and a Lipschitzian mapping \((x(.), z(.)): [0,r] \to \text{graph} u(.)\) such that: \((x(0), z(0)) = (x_ 0,z_ 0)\), \(z'(t) = \langle x'(t)\), \(p \rangle - F(x(t), z(t),p)\) a.e. in \(([0,r])\), suggests its interpretation as a generalization of the classical solutions obtained by Cauchy’s method of characteristics.

An equivalent definition, according to which a continuous function \(u(.)\) is a minimax solution iff its graph is weakly invariant with respect to the family of differential inclusions: \((x',z') \in E(x,z,p)\), \(p \in \mathbb{R}^ n\), where \(E(x,z,p) : = \{(v,w) \in \mathbb{R}^ n \times \mathbb{R}\); \(\| v \| \leq \rho (x,z)\), \(w= \langle v,p \rangle - F(x,z,p)\}\), leads to other equivalent definitions expressed in terms of tangential criteria for weakly invariant sets with respect to convex-valued differential inclusions.

The main result of Chapter I is Theorem 4.3 containing 7 equivalent definitions of the minimax solutions, in addition to those already mentioned, being the celebrated viscosity solutions for the equation \[ - F(x,u(x),Du(x)) = 0, \qquad x \in G \tag \(1'\) \] [e.g. M. G. Crandall and P.-L. Lions, Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)], and “contingent solutions”, defined in terms of differential inequalities verified by the extreme contingent directional derivatives of the solution.

The basic results, concerning uniqueness, existence, Hopf’s formulas and numerical methods for minimax solutions, are presented in Ch. II for Cauchy problems of the form: \[ D_ tu + H(t,x,u,D_ xu) = 0,\;(t,x) \in G : = (0, \theta) \times \mathbb{R}^ n, \quad u(\theta,x) = u_ 0(x)\;\forall x \in \mathbb{R}^ n, \tag{2} \] and in Ch. IV for Cauchy problems with additional conditions of the form: \(u(t,x) \leq w(t,x)\) \(\forall (t,x) \in G\), and also for Dirichlet problems of the form: \[ H(x,Du) - u = 0, \quad x \in G, \qquad u(x) = u_ 0(x)\quad \forall x \in \partial G. \tag{3} \] A large part of the book (Ch. III and part of Ch. IV) contains basic concepts and results from the theory of feedback (positional) differential games which is at the origin of the author’s concept of minimax and contingent solutions; in fact, the results of N. N. Krasovskij and the author published in the early 70s [e.g., Dokl. Akad. Nauk SSSR 190, 523-526 (1970; Zbl 0213.159), Positional differential games (in Russian) (Nauka 1974, Zbl 0298.90067)], concerning \(u\)-stable and \(v\)-stable solutions of the Bellman-Isaacs equation, anticipated with more than 10 years the theory of viscosity (super- and sub-) solutions.

The book also contains a 28 page Appendix presenting the necessary notations, definitions and auxiliary results from nonsmooth analysis and differential inclusions, and a large bibliographical list with 231 items.

In a brief but, of course, incomplete characterization, one may say that the present book provides several new equivalent definitions, new insight and an apparently more constructive approach to the theory of viscosity solutions and their applications to differential games.

Unfortunately, as in many other books and papers, any possible limits of the elaborated theory are either ignored or overlooked and lightly passed by; in this case (as in that of the theory of viscosity solutions) certain limits of the theory may be produced by the fact that the effective domain of the function \(F(.,.,.)\) in problem (1) is the set \(G \times \mathbb{R} \times \mathbb{R}^ n\). To illustrate these limits we give only two classical examples: as it is well known [e.g. L. Cesari, Optimization – theory and applications (Springer, N. Y., 1983; Zbl 0506.49001), the value function of the celebrated Brachistochrone problem is a classical (hence viscosity, minimax, etc.) solution of the problem \[ H(x, Du(x))=0, \quad x \in G \subset \mathbb{R}^ 2, \qquad u(x) = u_ 0(x) \quad \forall x \in G_ 0 \tag{4} \] defined by: \(G_ 0 = \{(0,0)\}\), \(u_ 0(0,0) = 0\), \(G = (0, \infty) \times (0, \infty)\), \(H(x,p) = 0\) if \(x = (x_ 1, x_ 2) \in G\), \(p \in \mathbb{R}^ 2\), \(\| p \| \leq 1/(x_ 2 + k)^{1/2}\) and \(H(x,p) = - \infty\) otherwise, where \(k \geq 0\) is a given constant; similarly, the value function of Plateau’s problem for surfaces of revolution is a locally-Lipschitz (viscosity, minimax, contingent) solution of the problem (4) defined by: \(G_ 0 = \{(0,y)\}\), \(y > 0\), \(u_ 0(0,y) = 0\), \(G = (0, \infty) \times [0, \infty)\), \(H(x,p) = 0\) if \(x = (x_ 1,x_ 2) \in G\), \(p \in \mathbb{R}^ 2\), \(\| p \| \leq x_ 2\) and \(H(x,p) = - \infty\) otherwise. Obviously, the two Hamiltonians above do not satisfy the hypotheses under which the results in the book are obtained; in particualar, problem (4) does not enjoy the uniqueness property since \(u(x) \equiv 0\) is a classical solution in both cases.

The first definition of the author’s “minimax solution”, as a continuous function, \(u(.) : G \to \mathbb{R}\), such that for any \((x_ 0,z_ 0) \in \text{graph} u(.)\) and any \(p \in \mathbb{R}^ n\) there exist \(r>0\) and a Lipschitzian mapping \((x(.), z(.)): [0,r] \to \text{graph} u(.)\) such that: \((x(0), z(0)) = (x_ 0,z_ 0)\), \(z'(t) = \langle x'(t)\), \(p \rangle - F(x(t), z(t),p)\) a.e. in \(([0,r])\), suggests its interpretation as a generalization of the classical solutions obtained by Cauchy’s method of characteristics.

An equivalent definition, according to which a continuous function \(u(.)\) is a minimax solution iff its graph is weakly invariant with respect to the family of differential inclusions: \((x',z') \in E(x,z,p)\), \(p \in \mathbb{R}^ n\), where \(E(x,z,p) : = \{(v,w) \in \mathbb{R}^ n \times \mathbb{R}\); \(\| v \| \leq \rho (x,z)\), \(w= \langle v,p \rangle - F(x,z,p)\}\), leads to other equivalent definitions expressed in terms of tangential criteria for weakly invariant sets with respect to convex-valued differential inclusions.

The main result of Chapter I is Theorem 4.3 containing 7 equivalent definitions of the minimax solutions, in addition to those already mentioned, being the celebrated viscosity solutions for the equation \[ - F(x,u(x),Du(x)) = 0, \qquad x \in G \tag \(1'\) \] [e.g. M. G. Crandall and P.-L. Lions, Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)], and “contingent solutions”, defined in terms of differential inequalities verified by the extreme contingent directional derivatives of the solution.

The basic results, concerning uniqueness, existence, Hopf’s formulas and numerical methods for minimax solutions, are presented in Ch. II for Cauchy problems of the form: \[ D_ tu + H(t,x,u,D_ xu) = 0,\;(t,x) \in G : = (0, \theta) \times \mathbb{R}^ n, \quad u(\theta,x) = u_ 0(x)\;\forall x \in \mathbb{R}^ n, \tag{2} \] and in Ch. IV for Cauchy problems with additional conditions of the form: \(u(t,x) \leq w(t,x)\) \(\forall (t,x) \in G\), and also for Dirichlet problems of the form: \[ H(x,Du) - u = 0, \quad x \in G, \qquad u(x) = u_ 0(x)\quad \forall x \in \partial G. \tag{3} \] A large part of the book (Ch. III and part of Ch. IV) contains basic concepts and results from the theory of feedback (positional) differential games which is at the origin of the author’s concept of minimax and contingent solutions; in fact, the results of N. N. Krasovskij and the author published in the early 70s [e.g., Dokl. Akad. Nauk SSSR 190, 523-526 (1970; Zbl 0213.159), Positional differential games (in Russian) (Nauka 1974, Zbl 0298.90067)], concerning \(u\)-stable and \(v\)-stable solutions of the Bellman-Isaacs equation, anticipated with more than 10 years the theory of viscosity (super- and sub-) solutions.

The book also contains a 28 page Appendix presenting the necessary notations, definitions and auxiliary results from nonsmooth analysis and differential inclusions, and a large bibliographical list with 231 items.

In a brief but, of course, incomplete characterization, one may say that the present book provides several new equivalent definitions, new insight and an apparently more constructive approach to the theory of viscosity solutions and their applications to differential games.

Unfortunately, as in many other books and papers, any possible limits of the elaborated theory are either ignored or overlooked and lightly passed by; in this case (as in that of the theory of viscosity solutions) certain limits of the theory may be produced by the fact that the effective domain of the function \(F(.,.,.)\) in problem (1) is the set \(G \times \mathbb{R} \times \mathbb{R}^ n\). To illustrate these limits we give only two classical examples: as it is well known [e.g. L. Cesari, Optimization – theory and applications (Springer, N. Y., 1983; Zbl 0506.49001), the value function of the celebrated Brachistochrone problem is a classical (hence viscosity, minimax, etc.) solution of the problem \[ H(x, Du(x))=0, \quad x \in G \subset \mathbb{R}^ 2, \qquad u(x) = u_ 0(x) \quad \forall x \in G_ 0 \tag{4} \] defined by: \(G_ 0 = \{(0,0)\}\), \(u_ 0(0,0) = 0\), \(G = (0, \infty) \times (0, \infty)\), \(H(x,p) = 0\) if \(x = (x_ 1, x_ 2) \in G\), \(p \in \mathbb{R}^ 2\), \(\| p \| \leq 1/(x_ 2 + k)^{1/2}\) and \(H(x,p) = - \infty\) otherwise, where \(k \geq 0\) is a given constant; similarly, the value function of Plateau’s problem for surfaces of revolution is a locally-Lipschitz (viscosity, minimax, contingent) solution of the problem (4) defined by: \(G_ 0 = \{(0,y)\}\), \(y > 0\), \(u_ 0(0,y) = 0\), \(G = (0, \infty) \times [0, \infty)\), \(H(x,p) = 0\) if \(x = (x_ 1,x_ 2) \in G\), \(p \in \mathbb{R}^ 2\), \(\| p \| \leq x_ 2\) and \(H(x,p) = - \infty\) otherwise. Obviously, the two Hamiltonians above do not satisfy the hypotheses under which the results in the book are obtained; in particualar, problem (4) does not enjoy the uniqueness property since \(u(x) \equiv 0\) is a classical solution in both cases.

Reviewer: S.Mirica (Bucureşti)

### MSC:

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

35F20 | Nonlinear first-order PDEs |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |

49N70 | Differential games and control |

49N75 | Pursuit and evasion games |