3
$\begingroup$

Something I've noticed while browsing book stores/libraries is that it is remarkably more difficult to find literature on parabolic PDEs than elliptic. For example check Amazon:

Parabolic

Elliptic

226 vs 427 results. And for good measure, hyperbolic:

Hyperbolic

351 results

Why is there such a huge discrepancy? Why are there almost double the amount of elliptic texts than parabolic? Both types have very important examples. Are elliptic easier to write about or more interesting? Is there more theory? Why the difference? Also why are hyperbolic in the middle?

$\endgroup$
3
  • 2
    $\begingroup$ When you look at a parabolic PDE, the first thing you want to know is the steady state. So you equal the derivative with respect to time to zero. And get elliptic problem. $\endgroup$
    – Artem
    Jul 1, 2017 at 15:11
  • $\begingroup$ @artem good point. Parabolic do reduce to elliptic. $\endgroup$
    – user223391
    Jul 1, 2017 at 15:21
  • $\begingroup$ Elliptic PDE arise frequently in geometry. (Parabolic PDE, then, tend to arise as ways to flow towards solutions of elliptic PDE we like.) $\endgroup$
    – user98602
    Jul 1, 2017 at 17:05

2 Answers 2

Reset to default
1
$\begingroup$

Elliptic equation are easier to solve than parabolic or hyperbolic equations. Besides that, the methods and solutions for elliptic equations give hints about methods, solutions and behavior of parabolic and hyperbolic equations.

Regularity results obtained in elliptic equations are used in order to obtain regularity of parabolic equations.

For instance, the Theory of Semigroup is used to solve the following type of parabolic problems $$\begin{cases} u_t+Au=f(f)\\ u(0)=u_0, \end{cases} $$ where $A$ is an unbounded linear operator. If this operator is the laplacian, i.e., $A=\Delta$, we have the heat equation. The Semigroup Theory can solve this problem easily, but to obtain regularity results we need developed the regularity theory for the laplacian equaiton, i.e., $$\Delta u=f.$$

$\endgroup$
1
$\begingroup$

I would argue that 427 vs 351 is not a "huge" discrepancy. In other words, there is insufficient evidence to support your claim.

$\endgroup$

You must log in to answer this question.