Chapter VII. Aspects of Partial Differential Equations

Abstract

This chapter provides an introduction to partial differential equations, particularly linear ones, beyond the material on separation of variables in Chapter I. Sections 1–2 give an overview. Section 1 addresses the question of how many side conditions to impose in order to get local existence and uniqueness of solutions at the same time. The Cauchy–Kovalevskaya Theorem is stated precisely for first-order systems in standard form and for single equations of order greater than one. When the system or single equation is linear with constant coefficients and entire holomorphic data, the local holomorphic solutions extend to global holomorphic solutions. Section 2 comments on some tools that are used in the subject, particularly for linear equations, and it gives some definitions and establishes notation. Section 3 establishes the basic theorem that a constant-coefficient linear partial differential equation $Lu = f$ has local solutions, the technique being multiple Fourier series. Section 4 proves a maximum principle for solutions of second-order linear elliptic equations $Lu = 0$ with continuous real-valued coefficients under the assumption that $L(1) = 0$. Section 5 proves that any linear elliptic equation $Lu = f$ with constant coefficients has a “parametrix,” and it shows how to deduce from the existence of the parametrix the fact that the solutions $u$ are as regular as the data $f$. The section also deduces a global existence theorem when $f$ is compactly supported; this result uses the existence of the parametrix and the constant-coefficient version of the Cauchy–Kovalevskaya Theorem. Section 6 gives a brief introduction to pseudodifferential operators, concentrating on what is needed to obtain a parametrix for any linear elliptic equation with smooth variable coefficients.