Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 9

Well-posed Problems

What happens if we consider a partial differential equation in a framework which
is meant for a different type?
To answer this question, we first turn to the wave equation (1.6), and attempt
to solve the boundary-value problem in the domain

2

={ (x, t) ∈ R ; a 1 <x + t<a 2 ,b 1 <x − t<b 2 } .

Here is a rotated rectangle, and its sides are parallel to the coordinate axes ξ,η
defined in (1.8). In view of u(ξ, η) = φ(ξ) + ψ(η) , the values of u on opposite
sides of can differ only by a constant. Thus, the boundary-value problem with
general data is not solvable. This also follows for differently shaped domains by
similar but somewhat more involved considerations.

2

}
as an initial-value problem , where y plays the role of time. Let n> 0. Assuming

{ (x, y) ∈ R ; y ≥

Next we study the potential equation (1.1) in the domain 0

1
n sin nx,

u(x, 0 ) =

u y (x, 0 ) = 0 ,

we clearly get the formal solution

1
n cosh ny sin nx,

u(x, y) =

which grows like e ny . Since n can be arbitrarily large, we draw the following
conclusion: there exist arbitrarily small initial values for which the corresponding
solution at y =

1is arbitrarily large . This means that solutions of this problem,
when they exist, are not stable with respect to perturbations of the initial values.
Using the same arguments, it is immediately clear from (1.13) that a solution
of a parabolic equation is well-behaved for t>t 0 , but not for t<t 0 . However,
sometimes we want to solve the heat equation in the backwards direction, e.g.,
in order to find out what initial temperature distribution is needed in order to get
a prescribed distribution at a later time t 1 > 0. This is a well-known improperly
posed problem. By (1.13), we can prescribe at most the low frequency terms of
the temperature at time t 1 , but by no means the high frequency ones.
Considerations of this type led Hadamard [1932] to consider the solvability
of differential equations (and similarly structured problems) together with the
stability of the solution.