The method of Newton’s polyhedron in the theory of partial differential equations.

*(English)*Zbl 0779.35001The authors develop Newton’s polyhedron method for some problems in the theory of partial differential equations. It is the study of a purely algebraic question which is associated with local solvability and local smoothness of solutions to partial differential equations. The book splits into two parts. Chapters 1 to 4 and Chapters 5 to 7 where Newton’s polygon and Newton’s polyhedron are considered.

In Chapter 1, the basic notions and constructions are presented for polynomials in relation to Newton’s polygon \(N(P)\) which is the closure of the set of such pairs \((\alpha',\beta')\) satisfying \(0\leq\alpha'< \alpha\), \(0\leq\beta'<\beta\), \((\alpha,\beta)\in \text{conv}(\nu(P))\), where \(\nu(P)\) is the set of all exponents of the polynomial \(P(\xi,\eta)\) under study. The main result is that the necessary and sufficient conditions for the validity of the inequality (1) \(|\xi^ \alpha \eta^ \beta|\leq c| P(\xi,\eta)|\) \(\forall(\alpha,\beta)\in\nu(P)\), \(\xi^ 2+\eta^ 2>c_ 0\), for a polynomial (2) \(P(\xi,\eta)= \sum_{(\alpha,\beta)\in \nu(P)} a_{\alpha\beta} \xi^ \alpha \eta^ \beta\), is that all quasi- homogeneous parts of \(P(\xi,\eta)\) have no zeros. The \(N\) quasi-elliptic polynomials and the differential equations having them as symbols are considered.

Chapter 2 is devoted to studying Cauchy’s problem for parabolic differential operators in several variables. Instead of inequality (1), the following inequality is analyzed in this chapter \[ \sum_{(\alpha,\beta)\in \Delta(P)} |\xi|^ \alpha |\tau|^ \beta< c| P(\xi,\tau)|, \qquad \text{Im } \tau\leq\gamma_ 0, \quad (\xi_ 1,\dots,\xi_ n,\text{Re }\tau) \in\mathbb{R}^{n+1}, \] where \(\Delta(P)\) coincides with Newton’s polynomial \(N(P)\) for \(n=1\). In terms of quasi-homogeneous parts, a new class of parabolic operators, namely the notion of \(N\)-parabolic and \(N\)-stable correct, which relates to classical operators parabolic in Petrovskij’s sense is described.

Dominantly correct operators are described in Chapter 3. It is concerned with inequalities of the following type \[ \sum_{(\kappa,\beta)\in \delta(P)} |\xi|^ \kappa |\tau|^ \beta \leq \varepsilon (\text{Im } \tau)| P(\xi,\tau)|, \qquad \varepsilon(\text{Im }\tau) \to 0, \quad \text{Im }\tau\to -\infty, \] where \(\delta(P)\) is the convex polygon spanned to the minor points of \(N(P)\). An existence and uniqueness theorem is proved for the solution of Cauchy’s problem in the spaces \(H^{(s)}_{[\gamma]}\).

Chapter 4 is devoted to the study of operators of principal type associated with Newton’s polygon, or differential operators \(P(x,y;D_ x,D_ y)\) such that for any bounded region \(\Omega\) of a sufficiently small diameter \(\text{diam }\Omega\leq \lambda<\lambda_ 0\) the inequality \[ \sum_{(\alpha,\beta)\in\delta(P)} \| D_ x^ \alpha D_ y^ \beta u\|\leq \varepsilon(\lambda)\| P(x,y;D_ x,D_ y)u\| \qquad \forall u\in {\mathcal D}(\Omega),\;\varepsilon(\lambda)\to 0,\;\lambda\to 0, \] holds. This leads, in the case of constant coefficients, to an algebraic condition \[ \sum_{(\alpha,\beta)\in\delta(P)} | \xi^ \alpha \eta^ \beta|\leq c(\widetilde{P+Q})(\xi,\eta) \qquad \forall Q\in {\mathcal L}_{N(P)} \] where \({\mathcal L}_{N(P)}\) is the class of those polynomials \(Q\) such that \(N(Q)\subset \delta(P)\). For the case of variable coefficients, additional conditions should be imposed in order to achieve the local solvability.

The results of this chapter are a generalization and further development of the well-known result by HĂ¶rmander on polynomials of principal type.

Chapters 5, 6, and 7 are devoted to passing from Newton’s polygon to Newton’s polyhedron.

In Chapter 1, the basic notions and constructions are presented for polynomials in relation to Newton’s polygon \(N(P)\) which is the closure of the set of such pairs \((\alpha',\beta')\) satisfying \(0\leq\alpha'< \alpha\), \(0\leq\beta'<\beta\), \((\alpha,\beta)\in \text{conv}(\nu(P))\), where \(\nu(P)\) is the set of all exponents of the polynomial \(P(\xi,\eta)\) under study. The main result is that the necessary and sufficient conditions for the validity of the inequality (1) \(|\xi^ \alpha \eta^ \beta|\leq c| P(\xi,\eta)|\) \(\forall(\alpha,\beta)\in\nu(P)\), \(\xi^ 2+\eta^ 2>c_ 0\), for a polynomial (2) \(P(\xi,\eta)= \sum_{(\alpha,\beta)\in \nu(P)} a_{\alpha\beta} \xi^ \alpha \eta^ \beta\), is that all quasi- homogeneous parts of \(P(\xi,\eta)\) have no zeros. The \(N\) quasi-elliptic polynomials and the differential equations having them as symbols are considered.

Chapter 2 is devoted to studying Cauchy’s problem for parabolic differential operators in several variables. Instead of inequality (1), the following inequality is analyzed in this chapter \[ \sum_{(\alpha,\beta)\in \Delta(P)} |\xi|^ \alpha |\tau|^ \beta< c| P(\xi,\tau)|, \qquad \text{Im } \tau\leq\gamma_ 0, \quad (\xi_ 1,\dots,\xi_ n,\text{Re }\tau) \in\mathbb{R}^{n+1}, \] where \(\Delta(P)\) coincides with Newton’s polynomial \(N(P)\) for \(n=1\). In terms of quasi-homogeneous parts, a new class of parabolic operators, namely the notion of \(N\)-parabolic and \(N\)-stable correct, which relates to classical operators parabolic in Petrovskij’s sense is described.

Dominantly correct operators are described in Chapter 3. It is concerned with inequalities of the following type \[ \sum_{(\kappa,\beta)\in \delta(P)} |\xi|^ \kappa |\tau|^ \beta \leq \varepsilon (\text{Im } \tau)| P(\xi,\tau)|, \qquad \varepsilon(\text{Im }\tau) \to 0, \quad \text{Im }\tau\to -\infty, \] where \(\delta(P)\) is the convex polygon spanned to the minor points of \(N(P)\). An existence and uniqueness theorem is proved for the solution of Cauchy’s problem in the spaces \(H^{(s)}_{[\gamma]}\).

Chapter 4 is devoted to the study of operators of principal type associated with Newton’s polygon, or differential operators \(P(x,y;D_ x,D_ y)\) such that for any bounded region \(\Omega\) of a sufficiently small diameter \(\text{diam }\Omega\leq \lambda<\lambda_ 0\) the inequality \[ \sum_{(\alpha,\beta)\in\delta(P)} \| D_ x^ \alpha D_ y^ \beta u\|\leq \varepsilon(\lambda)\| P(x,y;D_ x,D_ y)u\| \qquad \forall u\in {\mathcal D}(\Omega),\;\varepsilon(\lambda)\to 0,\;\lambda\to 0, \] holds. This leads, in the case of constant coefficients, to an algebraic condition \[ \sum_{(\alpha,\beta)\in\delta(P)} | \xi^ \alpha \eta^ \beta|\leq c(\widetilde{P+Q})(\xi,\eta) \qquad \forall Q\in {\mathcal L}_{N(P)} \] where \({\mathcal L}_{N(P)}\) is the class of those polynomials \(Q\) such that \(N(Q)\subset \delta(P)\). For the case of variable coefficients, additional conditions should be imposed in order to achieve the local solvability.

The results of this chapter are a generalization and further development of the well-known result by HĂ¶rmander on polynomials of principal type.

Chapters 5, 6, and 7 are devoted to passing from Newton’s polygon to Newton’s polyhedron.

Reviewer: H.Ding (Beijing)

##### MSC:

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

35A07 | Local existence and uniqueness theorems (PDE) (MSC2000) |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |

12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |

35E20 | General theory of PDEs and systems of PDEs with constant coefficients |

35J99 | Elliptic equations and elliptic systems |

35K99 | Parabolic equations and parabolic systems |