Chapter 6 Partial Differential Equations

4 CHAPTER 6. PARTIAL DIFFERENTIAL EQUATIONS
Linear partial differential equation’s are often classified as being of elliptic, hyperbolic, or
parabolic type. What follows is a brief and heuristic discussion of some of the features that
characterize these classifications of partial differential equations. Most of what is presented
in this introduction will be made precise in subsequent chapters.
Elliptic Equations
An example of an elliptic partial differential equation is Laplace’s equation,
div grad u =0
If u = u(x 1 , ··· ,x n ) and (x 1 , ··· ,x n ) are rectangular Cartesian coordinates, and w(x) =
(w 1 , ··· ,w n ),
grad u = ∇u =(u x1 , ··· ,u xn )
div ⃗w = ∂w + ···+ ∂w
∂x 1 ∂x n
and so
div grad u = ∇·∇u = ∇ 2 u =∆u = u x1 x 1
+ ···+ u xnx n
=0.
The Laplacian of u, ∆u, provides a comparison of values of u at a point with values at
neighboring points. To illustrate this idea consider the simplest case that u = u(x) and
assume u xx (x) > 0. Let ũ denote the tangent line approximation to u. For h>0 and
sufficiently small,
u(x + h) > ũ(x + h) =u(x)+u ′ (x)h
u(x − h) > ũ(x − h) =u(x) − u ′ (x)h
and so
u(x + h)+u(x − h)
>u(x).
2
Roughly stated, u(x) is smaller than its average value at nearby points if u xx (x) > 0. In
higher dimensions, say n = 2, the analogous statement is that if ∆u(x, y) > 0, then the
average value of u at neighboring points, say on a circle about (x, y), is greater than u(x, y).
If ∆u(x, y) < 0, then u(x, y) is greater than the average value of u on a circle about (x, y).
If ∆u(x, y) = 0 then u(x, y) is equal to its average value on a circle about (x, y). (We
will subsequently prove a theorem that makes these ideas rigorous.) More generally, if
∆u(x) =0,x∈ Ω ⊂ R n , then u is equal to its average value at neighboring points everywhere
in Ω. In a certain sense, this says that if u satisfies Laplace’s equation then u represents a
state of equilibrium.