0Introduction
III Analysis of Partial Differential Equations
0 Introduction
Partial differential equations are ubiquitous in mathematics, physics, and beyond.
The first equation we have met might be Laplace’s equation, saying
−∆u = −
n
X
i=1
∂
2
u
∂x
2
i
= 0.
This is the canonical example of an elliptic PDE, and we will spend a lot of
time thinking about elliptic PDEs, since they tend to be very well-behaved.
Instead of trying to explicitly solve equations, as we did in, say, IB Methods, our
focus is mostly on the existence and uniqueness of solutions, without explicitly
constructing them. This will involve the use of machinery from functional
analysis, and indeed a lot of the work will be about showing that we satisfy the
hypotheses required by the functional-analytic results (as well as proving the
functional-analytic results themselves (sometimes)).
We will also consider hyperbolic equations. The canonical example is the
wave equation
∂
2
u
∂t
2
− ∆u = 0.
The difference is that the time derivative term now has a different sign from the
rest. In Laplace’s equation, all directions were equal. Here time is a “special”
direction, and often our questions are about how the solution evolves in time.
Of course, we don’t “just solve” such equations. Usually, we impose some
data, such as the desired values of
u
on the boundary of our domain, or the
“starting configuration” in the case of the wave equation. In general, given such
a system, there are several questions we can ask:
– Does a solution exist?
– Is the solution unique?
– Does the solution depend continuously on the data?
–
How regular is the solution? Is it continuously differentiable? Or even
smooth?
These questions are closely related. To even make sense of the question, we
need to specify our “search space”, i.e. the sort of functions we are willing to
consider. For example, we may consider the space of all smooth functions, or
less ambitiously, the space of all twice-differentiable functions. This somewhat
answers the last question, but it doesn’t answer it completely. It could be that
we can try to search for the solution in the space of
C
2
functions, but it turns
out the solutions are always smooth!
The choice of this function space affects the answers to the other questions as
well. If we have a larger function space, then we are more likely to get a positive
answer to the first question. However, since there are more functions around, we
are more likely to get a negative function to the second question. So there is
some tension here.
The choice affects the third question in a slightly more subtle way. To speak
of continuity, we must pick a topology, and this usually comes from a norm on
the function space. Thus, to make sense of the third question, we must pick the
appropriate norm on both the space of data and the space of potential solutions.
After choosing the appropriate function spaces, if the answers to the first
three questions are all “yes”, then we say the problem is well-posed .