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Results tagged with ap.analysis-of-pdes
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user 4362
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
53
votes
Motivation for and history of pseudo-differential operators
I don't know the history at all, but I have to imagine that the language was invented to provide a context for talking about solution operators for differential equations. Consider, for example, the …
- 29.2k
24
votes
3
answers
3k
views
Does elliptic regularity guarantee analytic solutions?
Let $D$ be an elliptic operator on $\mathbb{R}^n$ with real analytic coefficients. Must its solutions also be real analytic? If not, are there any helpful supplementary assumptions? Standard Sobole …
- 29.2k
15
votes
Accepted
The principal symbol as an element in the K-theory
It's a bit easier to see this using a slightly non-standard definition of topological K-theory. Given a locally compact Hausdorff space $X$, let $\bf{E}$ be a complex of vector bundles, i.e. a sequenc …
- 29.2k
12
votes
Why is the harmonic oscillator so important? (pure viewpoint sought). How to motivate its ro...
I too am but a mere graduate student trying to sort through some of these same issues, but I might have some helpful insight. I'll let you be the judge.
The basic idea behind the heat equation proof …
- 29.2k
10
votes
Elliptic operators corresponds to non vanishing vector fields
Perhaps you would be interested in Witten's proof of the Poincare-Hopf theorem. Given a smooth nondegenerate vector field $V$ on a smooth closed manifold $M$, the theorem asserts that the Euler chara …
- 29.2k
3
votes
Spectral multipliers vis-a-vis Differential geometry
One of the major geometric applications of the sort of analysis that you describe is to index theory for elliptic operators on manifolds. Using geometry one can often construct a differential operato …
- 29.2k