## New answers tagged differential-operators

3
votes

Accepted

### Question about differential operators in a completely non-integrable distribution

Consider the following example: On $\mathbb{R}^4$ with coordinates $u^1,u^2,u^3, z^1$, define $z^2 = u^2 - z^1 u^1$ and $z^3 = u^3 - z^1u^2$.
We have that $U\cap Z$ is spanned by $\mathrm{d}u^2-z^1\,\...

5
votes

Accepted

### Why are we interested in spectral gaps for Laplacian operators

A spectral gap gives information on geometry of the manifold via Cheeger's inequality, https://en.wikipedia.org/wiki/Cheeger_constant See also Buser's inequality discussed there. More directly, a ...

2
votes

Accepted

### Harmonic polynomials on the sphere

I view this as a concatenation of two facts:
Fact 1: Let $k$ be a field, let $I$ be an ideal of $k[x_1, \ldots, x_n]$ and let $J$ be associated graded ideal of $I$, meaning that a degree $d$ ...

5
votes

### Harmonic polynomials on the sphere

There is a similar separation of the variables for ellipsoids
$x^2/a^2+y^2/b^2+z^2/c^2=1$ related to ellipsoidal harmonic functions,
see Whittaker Watson, Chap. 23. There is a generalization to
...

Top 50 recent answers are included

#### Related Tags

differential-operators × 458dg.differential-geometry × 149

fa.functional-analysis × 95

ap.analysis-of-pdes × 80

differential-equations × 65

operator-theory × 45

sp.spectral-theory × 43

riemannian-geometry × 42

elliptic-pde × 42

reference-request × 33

ag.algebraic-geometry × 32

index-theory × 28

pseudo-differential-operators × 25

ca.classical-analysis-and-odes × 21

d-modules × 17

real-analysis × 15

ds.dynamical-systems × 15

laplacian × 15

linear-pde × 14

rt.representation-theory × 13

ra.rings-and-algebras × 13

mp.mathematical-physics × 13

vector-bundles × 13

at.algebraic-topology × 12

lie-algebras × 12