Q. Which high-degree derivatives play an essential role
in applications, or in theorems?
Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the 3rd derivative (jolt or jerk) certainly play several roles in applications.
And the torsion of a curve in $\mathbb{R}^3$ can be expressed
using 3rd derivatives.
Beyond this, I'm out of my depth of experience. I know of the biharmonic equation
$\nabla ^4 \phi=0$.
There is a literature on the solvability of quintics,
but it seems this work is neither aimed at applications nor essential to
further theoretic developments.
(I am happy to have my ignorance corrected here.)
Q. What are examples of applications that depend on 4th-derivatives
(snap/jounce) or higher?
Are there substantive theorems that require existence of $\partial^4$ or higher as
assumptions, but do not require (or are not known to require) smoothness—derivatives of all orders?
Best Answer
Given two sets $A$ and $B$ in $\mathbb{R}^n$, the Minkowski sum written $A+B$ is the set $\{a+b:a\in A,b\in B\}$.
If $A$ and $B$ are convex subsets of $\mathbb{R}^2$ with real-analytic boundaries then the boundary of $A+B$ is only guaranteed to be '$6\frac{2}{3}$ times differentiable,' by which I mean $6$ times differentiable with $6$th derivative Hölder continuous with exponent $\frac{2}{3}$. This is known to be sharp.