0
votes
0answers
18 views

Vanishes of second order

I encountered the following statement in an article: Let $f(X) = f(X_1, ..., X_N) = \sum_{|\alpha| \geq 2} {a_{\alpha}X^{\alpha}}$ be a convergent power series which vanishes of second order at ...
0
votes
0answers
14 views

Deriving lower bound for eigenvalues of laplace operator

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 \quad \mbox{ on } \quad D, \quad u = 0 \quad \mbox{ on } \partial D $$ and let $w$ be a function such that $\Delta w + \beta w < 0$ ...
2
votes
0answers
57 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
8
votes
1answer
63 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
0
votes
1answer
28 views

Using Gauss's Theorem in weak formulation

Can anyone see how exactly Gauss's theorem used in the following case: In defining the weak solution to the linear elliptic equation we start with $$-\sum_{j,k=1}^{n}D_{j}(a_{jk}D_{k}u) + cu = f ...
0
votes
2answers
30 views

Green's Theorem and Divergence (2D)

I am reading the book Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson. In Chapter 1 he talks about the Possion Equation, and to prove that FEM ...
11
votes
1answer
117 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
5
votes
0answers
50 views

Inhomogeneous Wave Equation in 3 dimensions

From section 7.5 in this source, I see that, for $\vec{x} \in \mathbb{R}^3$, if $$ \frac{\partial^2 u\left(\vec{x},t\right)}{\partial t^2} - \nabla^2u\left(\vec{x},t\right) ...
7
votes
1answer
147 views

The proof of the Helmholtz decomposition theorem through Neumann boundary value problem

As you know, the Helmholtz decomposition theorem is as follows. Let $\Omega$ be open and simply connected, and bounded region in $\mathbb R^3$. Then the smooth vector field $E : \Omega \to ...
3
votes
1answer
97 views

Yang–Mills theory

We define the energy as $$E = I_F + I_K + I_V,$$ where, $$I_F [A]= \frac{1}{2} \int d^Dx \operatorname{tr} F^2_{ij},$$ $F_{ij}$ represents the electromagnetic force. $$I_K [\phi,A]= \frac{1}{2} \int ...
1
vote
1answer
152 views

Vector analysis: $(\vec v \cdot \vec \nabla) \vec v=(\vec \nabla \cdot \vec v) \vec v$?

If I know that $\vec \nabla \cdot \vec v=0$, can I say that: $$( \vec v \cdot \vec \nabla )\vec v=\underbrace{(\vec \nabla \cdot \vec v)}_{=0} \vec v=0 $$ ? Note: this is a question I asked in ...
4
votes
1answer
152 views

Divergence theorem in volume integral

We have a partial differential equation \begin{equation} \nabla \cdot (p_1^2\nabla\alpha)=0\,. \end{equation} Question: from this equation how can I write the following condition? \begin{equation} ...
4
votes
2answers
64 views

curl of what yields $(0,s^{-1},0)$ in cylindrical coordinates?

In cylindrical coordinates $(s,\theta,z)$, what function $\mathbf{A}$ has the property $$\nabla\times \mathbf{A} = (0, \frac{1}{s} , 0) $$ I know generally that $$\nabla\times \mathbf{A} = ...
2
votes
1answer
142 views

Dimensions analysis in Differential equation

Differential equation of solitary wave oscillons is defined by, $$ \Delta S -S +S^3=0 $$ How can we write this equation as, \begin{equation} \langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle ...
2
votes
1answer
134 views

Divergence free, smooth functions on unit circle.

I need to construct a divergence free, smooth, vector function on a unit circle such that $ \mathbf{u} = (u_1,u_2) = (0,0) $ on $\partial B$, and $\int\limits_B u_i \neq 0, \ i=1,2$. I was able to ...
1
vote
1answer
284 views

Solution for 3D wave equation for vector fields

I've been trying to find the solution to the 3-D wave equation for vector fields, that is: $$\nabla^2 \vec{u}(\vec{x}, t) = \frac{1}{c^2} \frac{\partial^2 \vec{u}(\vec{x}, t)}{\partial t^2}$$ Given ...
1
vote
1answer
69 views

Writing $\int_\Omega \Delta u \Delta v$ in a nicer way?

Is there a way to write $\int_\Omega (\Delta u)^2$ or more generally, $\int_\Omega \Delta u \Delta v$ more nicely (possibly after integrating by parts)? I want something like $\int \nabla f\cdot ...
3
votes
1answer
136 views

What is the name of this equation?

I have found this picture but I don't know the name of the equation in it. Another thing, what kind of plots are those in the picture? I have also tried to re copy it: $$ ...
1
vote
0answers
135 views

Fundamental solution of a vector field

Consider a smooth real vector field $X(x,\partial _x)=\sum a_i(x) \frac{\partial}{\partial x_i}$ on an open subset $\Omega \subset \mathbb{R}^n$ (which is assumed to be sufficiently regular) as a ...
3
votes
2answers
207 views

PDE involving curl

I need to solve the equation: $$\operatorname{curl} \left(\operatorname{curl}(\mathbf{u}(x,y,z)) \right)=\mathbf{v}(x,y,z)$$ Because $$\operatorname{curl}(\operatorname{curl}(\mathbf{f}))=-\nabla^2 ...
2
votes
1answer
111 views

Higher Dimensional Generalization of Helmholtz Theorem

We know that given the divergence and curl of a vector field (and appropriate boundary conditions) it is possible to construct a unique vector field in $\mathbb R^3$. The specific problem I am ...
3
votes
1answer
154 views

Is the problem asking to show that $r\times \nabla \psi$ satisfies wave equation wrong?

Considering the wave equation in spherical coordinates, if we know that $\psi(\vec{r})$ is a solution, then $\vec{r}\times \nabla \psi$ is also a solution. (The hint is to take the difference between ...