1.1 Partial diﬀerential equations
1.3 Strategies for studying PDE
This chapter surveys the principal theoretical issues concerning the solv-
ing of partial diﬀerential equations.
To follow the subsequent discussion, the reader should ﬁrst of all turn
to Appendix A and look over the notation presented there, particularly the
multiindex notation for partial derivatives.
1.1. PARTIAL DIFFERENTIAL EQUATIONS
A partial diﬀerential equation (PDE) is an equation involving an unknown
function of two or more variables and certain of its partial derivatives.
Using the notation explained in Appendix A, we can write out symbol-
ically a typical PDE, as follows. Fix an integer k ≥ 1 and let U denote an
open subset of
DEFINITION. An expression of the form
. . . , Du(x), u(x), x) = 0 (x ∈ U)
is called a kth-order partial diﬀerential equation, where
× · · · ×
× R × U → R