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

Abstract

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 in some 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.