The Method of Newton’s Polyhedron in the Theory of Partial Differential Equations

1. Two-sided estimates for polynomials related to Newton’s polygon and their application to studying local properties of partial differential operators in two variables.- §1. Newton’s polygon of a polynomial in two variables.- §2. Polynomials admitting of two-sided estimates.- §3. N Quasi-elliptic polynomials in two variables.- §4. N Quasi-elliptic differential operators.- Appendix to §4.- 2. Parabolic operators associated with Newton’s polygon.- §1. Polynomials correct in Petrovski?’s sense.- §2. Two-sided estimates for polynomials in two variables satisfying Petrovski?’s condition. N-parabolic polynomials.- §3. Cauchy’s problem for N-stable correct and N-parabolic differential operators in the case of one spatial variable.- §4. Stable-correct and parabolic polynomials in several variables.- §5. Cauchy’s problem for stable-correct differential operators with variable coefficients.- 3. Dominantly correct operators.- §1. Strictly hyperbolic operators.- §2. Dominantly correct polynomials in two variables.- §3. Dominantly correct differential operators with variable coefficients (the case of two variables).- §4. Dominantly correct polynomials and the corresponding differential operators (the case of several spatial variables).- 4. Operators of principal type associated with Newton’s polygon.- §1. Introduction. Operators of principal and quasi-principal type.- §2. Polynomials of N-principal type.- §3. The main L2 estimate for operators of N-principal type.- Appendix to §3.- §4. Local solvability of differential operators of N-principal type.- Appendix to §4.- 5. Two-sided estimates in several variables relating to Newton’s polyhedra.- §1. Estimates for polynomials in ?n relating to Newton’s polyhedra.- §2. Two-sided estimates insome regions in ?n relating to Newton’s polyhedron. Special classes of polynomials and differential operators in several variables.- 6. Operators of principal type associated with Newton’s polyhedron.- §1. Polynomials of N-principal type.- §2. Estimates for polynomials of N-principal type in regions of special form.- §3. The covering of ?n by special regions associated with Newton’s polyhedron.- §4. Differential operators of ?n-principal type with variable coefficients.- Appendix to §4.- 7. The method of energy estimates in Cauchy’s problem §1. Introduction. The functional scheme of the proof of the solvability of Cauchy’s problem.- §2. Sufficient conditions for the existence of energy estimates.- §3. An analysis of conditions for the existence of energy estimates.- §4. Cauchy’s problem for dominantly correct differential operators.- References.