1.4. OVERVIEW 11 study of eigenvalues, including a discussion of the principal eigenvalue for nonselfadjoint operators. Chapter 7 expands the energy methods to a variety of linear partial differential equations characterizing evolutions in time. We broaden our earlier investigation of the heat equation to general second-order parabolic PDE and of the wave equation to general second-order hyperbolic PDE. We study as well linear first-order hyperbolic systems, with the aim of motivat- ing the developments concerning nonlinear systems of conservation laws in Chapter 11. The concluding section 7.4 presents the alternative functional analytic method of semigroups for building solutions. (Missing from this long Part II on linear partial differential equations is any discussion of distribution theory or potential theory. These are impor- tant topics, but for our purposes seem dispensable, even in a book of such length. These omissions do not slow us up much and make room for more nonlinear theory.) PART III: Theory for Nonlinear Partial Differential Equations This section parallels for nonlinear PDE the development in Part II but is far less unified in its approach, as the various types of nonlinearity must be treated in quite different ways. Chapter 8 commences the general study of nonlinear partial differential equations with an extensive discussion of the calculus of variations. Here we set forth a careful derivation of the direct method for deducing the ex- istence of minimizers and discuss also a variety of variational systems and constrained problems, as well as minimax methods. Variational theory is the most useful and accessible of the methods for nonlinear PDE, and so this chapter is fundamental. Chapter 9 is, rather like Chapter 4 earlier, a gathering of assorted other techniques of use for nonlinear elliptic and parabolic partial differential equa- tions. We encounter here monotonicity and fixed point methods and a vari- ety of other devices, mostly involving the maximum principle. We study as well certain nice aspects of nonlinear semigroup theory, to complement the linear semigroup theory from Chapter 7. Chapter 10 is an introduction to the modern theory of Hamilton–Jacobi PDE and in particular to the notion of “viscosity solutions”. We encounter also the connections with the optimal control of ODE, through dynamic programming.

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