10 1. INTRODUCTION can, locally at least, be transformed into systems of ordinary differential equations (ODE), the characteristic equations. We stipulate that once the problem becomes “only” the question of integrating a system of ODE, it is in principle solved, sometimes quite explicitly. The derivation of the characteristic equations given in the text is very simple and does not require any geometric insights. It is in truth so easy to derive the characteristic equations that no real purpose is had by dealing with the quasilinear case first. We introduce also the Hopf–Lax formula for Hamilton–Jacobi equa- tions (§3.3) and the Lax–Oleinik formula for scalar conservation laws (§3.4). (Some knowledge of measure theory is useful here but is not essential.) These sections provide an early acquaintance with the global theory of these im- portant nonlinear PDE and so motivate the later Chapters 10 and 11. Chapter 4 is a grab bag of techniques for explicitly (or kind of explicitly) solving various linear and nonlinear partial differential equations, and the reader should study only whatever seems interesting. The section on the Fourier transform is, however, essential. The Cauchy–Kovalevskaya Theo- rem appears at the very end. Although this is basically the only general exis- tence theorem in the subject, and thus logically should perhaps be regarded as central, in practice these power series methods are not so prevalent. PART II: Theory for Linear Partial Differential Equations Next we abandon the search for explicit formulas and instead rely on functional analysis and relatively easy “energy” estimates to prove the ex- istence of weak solutions to various linear PDE. We investigate also the uniqueness and regularity of such solutions and deduce various other prop- erties. Chapter 5 is an introduction to Sobolev spaces, the proper setting for the study of many linear and nonlinear partial differential equations via en- ergy methods. This is a hard chapter, the real worth of which is only later revealed, and requires some basic knowledge of Lebesgue measure theory. However, the requirements are not really so great, and the review in Ap- pendix E should suffice. In my opinion there is no particular advantage in considering only the Sobolev spaces with exponent p = 2, and indeed in- sisting upon this obscures the two central inequalities, those of Gagliardo– Nirenberg–Sobolev (§5.6.1) and of Morrey (§5.6.2). In Chapter 6 we vastly generalize our knowledge of Laplace’s equation to other second-order elliptic equations. Here we work through a rather com- plete treatment of existence, uniqueness and regularity theory for solutions, including the maximum principle, and also a reasonable introduction to the
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