# Tagged Questions

**0**

votes

**0**answers

83 views

### Wave operator for focusing NLS

Consider the NLS equation
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u+u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\right.
\end{equation}
where ...

**4**

votes

**2**answers

171 views

### First order Elliptic operator

Assume that there exists a first order elliptic operator $D$ acting on functions from $\mathbb{R}^n$ to some vector space $V$. What can we conclude about $V$?
For example, is the dimension of $V$ ...

**1**

vote

**1**answer

237 views

### solving elliptic system of first-order linear PDE's

I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which ...

**7**

votes

**2**answers

238 views

### Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...

**1**

vote

**1**answer

177 views

### Reference to the Existence and Uniqueness of the PDE system

Hi all
I've the following Problem on systems of Partial Differential Equations.I have " N " Physical variables. and Finally I form the equation on a bounded domain having regular boundary in R^d. ...

**3**

votes

**1**answer

335 views

### Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...

**2**

votes

**2**answers

359 views

### How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?

To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...

**1**

vote

**0**answers

151 views

### On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta ...

**2**

votes

**0**answers

260 views

### A free boundary problem by finite difference method

I wanna discretize the following free boundary problem
Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.
I apply finite ...

**5**

votes

**1**answer

354 views

### A moving boundary in rock mechanics

I'm concern a moving boundary problem in rock mechanics.
We consider a problem of unsaturated flow of an in-compressible fluid in a
porous medium(rock) like D. Moreover suppose that support of a ...

**2**

votes

**0**answers

282 views

### Vanishing solution of the Poisson equation at infinity

Hi, I am interested in finding some vanish bahavior at infinity of the solutions of this kind of equations:
$-\Delta\phi+a(x)\phi=b(x)$
where $a(x), b(x)\in L^{p}$ with $1\leq p\leq 3$. Besides ...

**1**

vote

**1**answer

841 views

### Poisson equation with special Neumann BC

Hi
Consider Poisson equation with Neumann boundary condition but the right hand side of boundary condition is in term of the unknown function $u$.
How we can solve it?
$\Delta u(x) = f(x)\quad in~ ...