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