Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with ap.analysis-of-pdes
Search options questions only
not deleted
user 16183
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
10
votes
1
answer
812
views
Hodge decomposition in elliptic complexes
EDIT: In the book "Principles of Algebraic Geometry" by Griffiths and Harris the authors prove the Hodge decomposition for the Dolbeault operator $\bar\partial$ on differential forms on a compact comp …
- 21.8k
8
votes
2
answers
2k
views
Estimates on the Green function of an elliptic second order differential operator.
Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free ter …
- 21.8k
7
votes
1
answer
354
views
Higher regularity of solutions of non-linear elliptic PDE
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Let $u\in C^2(\bar \Omega)$ be a solution of the Dirichlet problem for the non-linear equation
\begin{eqnarray}
F( …
- 21.8k
7
votes
2
answers
3k
views
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist …
- 21.8k
4
votes
1
answer
535
views
Path integral presentation of solutions of Dirac equation
It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold).
Is there a way to present solutions of the Dirac equation using path …
- 21.8k
4
votes
1
answer
196
views
Existence of eigen basis for elliptic operator on compact manifold
Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting o …
- 21.8k
4
votes
1
answer
451
views
Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation
…
- 21.8k
3
votes
0
answers
86
views
Estimate a function given an estimate of its Laplacian
Let $f_\lambda\geq 0$ with $\lambda>0$, be smooth functions in the unit Euclidean ball $B\subset \mathbb{R}^n$ satisfying the following conditions:
\begin{eqnarray*}
\int_B |f_\lambda(x)|^2dx\leq 1,\\ …
- 21.8k
3
votes
1
answer
173
views
Solvability of general linear PDE with constant coefficients
Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.
Is it true that the equation $$Du=f$$
is solvable in any …
- 21.8k
3
votes
1
answer
2k
views
Boundary conditions for Klein-Gordon equation
Let us consider the Klein-Gordon equation
$$(\Box +m^2)u=0,$$
where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. …
- 21.8k
3
votes
3
answers
2k
views
Uniqueness of solution of the wave equation
Consider the wave equation
$$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$
with initial conditions
$$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$
Does …
- 21.8k
2
votes
1
answer
412
views
Second order estimates of Monge-Ampere equations
In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the …
- 21.8k
2
votes
2
answers
349
views
Estimate of a solution of Schroedinger equation for a free particle
Let $\psi(x,t)$ be a solution of the Schroedinger on the line
$$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$
One assumes that $\psi(x,0)$ "behaves well" as …
- 21.8k
2
votes
0
answers
163
views
Inhomogeneous wave equation - a reference
Consider the inhomogeneous wave equation
$$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$
where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\ …
- 21.8k
2
votes
2
answers
141
views
Equality of spectra of products of operators
Let $A$ be a linear operator between two Hilbert spaces. Let $A^*$ be its adjoint.
Question. Under what conditions the non-zero spectra of $A^*A$ and $AA^*$ coincide counting multiplicities?
In my s …
- 21.8k