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.