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Results tagged with ap.analysis-of-pdes
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user 26801
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
1
answer
377
views
Longtime behaviour of the periodic KdV equation
I was wondering if anyone could give a heuristic (i.e. preferably non-technical) explanation of what is the expected longtime behavior of the periodic KdV equation.
Recall the standard KdV equation …
- 2,299
1
vote
1
answer
79
views
Finding singular "solutions" to the Dirichlet problem for Schrödinger operators that do not ...
Suppose that $\Omega\subset \mathbb{R}^n$ is a smooth open region and that $V:\Omega\to \mathbb{R}^+$ be a positive smooth function.
Then we have a family of operators
$$L_\epsilon =-\Delta -\epsilon …
- 2,299
1
vote
Accepted
Partial $L^2$ control on (part of) the Hessian of a harmonic function.
Okay so I thought about this some more and I believe it is just a (really) straightforward application of the Poincare/Wirtinger inequality and the obvious way you would solve $u_{xy}=0$ classically. …
- 2,299
1
vote
1
answer
391
views
Partial $L^2$ control on (part of) the Hessian of a harmonic function.
I have a simple little analysis question that I'm hoping is well known.
Suppose $D=\lbrace(x,y): x^2+y^2<1\rbrace$ is the unit disk and that $u$ is a harmonic function on $D$. Suppose in addition th …
- 2,299
7
votes
PDEs as a tool in other domains in mathematics
How about Hodge theory. I.e. that each DeRham cohomology class of a smooth compact manifold has a harmonic representative (one has to of course choose a Riemannian metric to make sense of harmonic). T …
2
votes
0
answers
267
views
Finer properties of a sequence of harmonic functions
This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.
Background:
When o …
- 2,299
9
votes
Does Ricci flow with surgery come from sections of a smooth Riemannian manifold?
To build a little on what Agol said:
For mean curvature flow of hypersurfaces, the analogous question is at least partially known to be true (sort of). The advantage of the mean curvature flow over …
- 2,299
2
votes
1
answer
2k
views
The normal derivative of the Green's function
I was wondering if anything was known about the following:
Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk.
Consider now the Green's functions $G(z; p)$ of t …
- 2,299
3
votes
0
answers
318
views
Controlling the Second Eigenvalue of a Schrödinger Operator
Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$.
Let $L$ be the operator
$$
L=\Delta+V
$$
where $\Delta$ is the Laplace-beltrami operator on $M$ (so is …
- 2,299
3
votes
Accepted
Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds
The desired bound is correct (and in fact you get a fairly explicit value for $\epsilon$ of anything below $4\pi$) . A proof can be found in these beautiful notes of a course by Brian White (it's Th …
- 2,299