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Results tagged with ap.analysis-of-pdes
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user 613
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
6
votes
Accepted
Reference for $\epsilon$-regularity
First, it is straightforward to adapt Haslhofer's proof to higher dimensions, using the Sobolev inequality: For any compactly supported $u$ such that $\|\nabla u\|_2 < \infty$,
$$ \|u\|_{\frac{2n}{n-2 …
- 27.5k
3
votes
Accepted
Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term
Here's what I would try:
Check that for each $t$ and $f \in S$, where $S$ is an appropriately chosen function space, there is a unique solution to the elliptic problem:
$$ L\phi = f \text{ on }\Omega …
- 27.5k
8
votes
Accepted
Elliptic regularity on manifolds: Is this true?
Use a partition of unity to reduce the statement to a local one for a function compactly supported on a coordinate chart. At that point, any elliptic regularity theorem on an open domain in Rn can be …
- 27.5k
24
votes
Accepted
PDEs and algebraic varieties
What has been studied more extensively is the top order term, which is known as the principal symbol. Since it is a homogeneous polynomial of degree $d$, it defines a real projective algebraic variety …
- 27.5k
3
votes
Accepted
Reference request: inverse of differential operators
There are many possible answers to this question, depending on the type of the differential operator, the domain of the functions, and whether you want to impose any additional conditions such as spec …
- 27.5k
2
votes
Explicit solutions for linear system of PDEs with constant coefficients
ADDDED: Since equation (1) below is a first order linear ODE,it has an explicit solution (with an integral). It follows that the solution to the entire system of equations can be written explicitly as …
- 27.5k
15
votes
Applications of microlocal analysis?
Although microlocal analysis was developed originally exclusively for linear problems, it has played an increasingly important role in nonlinear PDE via what's known as paradifferential calculus. Ther …
- 4,713
21
votes
Does Ricci flow depend continuously on the initial metric?
If you use the right topology on the space of metrics, the answer is yes. Basically, this is always true and a consequence of the proof for any theorem on the existence, uniqueness, and regularity of …
- 4,713
3
votes
Accepted
How to prove the reverse Hölder inequality for Laplace equations?
The inequality is scale invariant and holds for a ball of any radius. It follows by a standard argument that is the inductive step in what's known as Moser iteration.
The constant $C$ below can change …
- 27.5k
3
votes
Accepted
Do Laplace-Beltrami eigenfunctions vary continuously with the metric?
EDITED: Added clarification, as pointed out by @TerryTao.
Let $g_1$, $g_2$ be Riemannian metrics and $\Delta_1$, $\Delta_2$ their respective Laplacians. Let $\lambda_1$ be an eigenvalue of $\Delta_1$ …
- 27.5k
1
vote
Local solvability and Cauchy-Kovalevskaya theorem for PDEs
The system you wrote down
First, let's assume everything is smooth.
\begin{align*}
u_t &= v\\
v_{tt} &= u_x\\
\end{align*}
is equivalent to the first order system
\begin{align*}
u_t &= v\\
v_t …
- 27.5k
4
votes
Accepted
Existence for an overdetermined system of PDEs
COMMENT: The answer below is just the proof of the Frobenius theorem (https://en.wikipedia.org/wiki/Frobenius_theorem_(differential_topology) applied to this specific case. The arguments below are als …
- 27.5k
6
votes
System of linear pde with non constant coefficients
If you switch the second and third rows of your system, the differential operator is the same as the linearization of equation (4.3) in Existence of elastic deformations with prescribed principal stra …
- 27.5k
1
vote
The Monge- Ampère equation with a non positive right hand side
In general nothing is known. Only local solvability is known for some special cases.
- 27.5k
6
votes
Accepted
Hyperbolic PDE in mathematics
Hyperbolic PDEs arise unexpectedly in some differential geometric questions involving prescribed data. What's weird in these cases is that there is no natural time coordinate in the PDEs. Here are som …
- 27.5k