1
vote
0answers
19 views

Method of PDE solution by Fourier transform

In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded ...
4
votes
2answers
98 views

Uniqueness for tempered distributional Cauchy problems

Question. Assume that $U\in C^1(\,[0, \infty)\to \mathcal{S}'(\mathbb{R}^n)\,)$ is a solution to the following tempered distributional Cauchy problem $$\tag{CP}\begin{cases} \frac{ d U}{dt} = f ...
0
votes
0answers
41 views

Infinite solutions of Navier-Stokes equations

Is it a known fact that Navier-Stokes equations have exactly one (possibly infinite) solution in the space of distributions?
2
votes
1answer
54 views

Weak Laplacian of $\|x\|^\alpha$

Let $\alpha> 0$ and consider the function $\|\mathbf x\|^\alpha = (x^2 + y^2)^{\frac{\alpha}{2}}$ defined on $\mathbb R^2$. I want to compute the Laplacian $\Delta (\|\mathbf x\|^\alpha)$ in the ...
0
votes
0answers
35 views

P.D.E's - Distribution

If we have $$\int_{\Omega} (u_m-u_n) \Delta \varphi \,dx=\int_{\Omega} |u_m-u_n|\,dx$$ for $\Delta\varphi = \operatorname{sgn}(u_m-u_n)$. Now, if I need to choose a test function $\varphi$ such that ...
1
vote
2answers
78 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
3
votes
1answer
62 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
1
vote
1answer
54 views

When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to ...
1
vote
1answer
95 views

Fundamental solution of nonlinear PDE

A fundamental solution of a linear PDE (in sense of Schwartz), $Lu=0$ is defined as a distribution $E$ such that $LE=\delta$. Now I wish to find fundamental solution of nonlinear PDE, such as the ...
8
votes
0answers
73 views

Delta distributions with nonlinear arguments

I am confused by the use of nonlinear arguments with the Dirac $\delta$ distribution that I am encountering in the literature. This looks like a widespread use, but for concreteness let us focus on a ...
2
votes
1answer
248 views

Fourier Transform of Dirac Comb on $\mathbb{Z}$ and $\mathbb{Z}^{d}$.

Let $f(x)=\sum_{n\in\mathbb{Z}}\delta(x-n).$ (a) Show $f$ is a tempered distribution. (b) Compute $\hat{f}$ using the convention $\int_{\mathbb{R}}f(x)e^{-ix\xi}\;dx$ convention for $\mathcal{F}$. ...
4
votes
0answers
65 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...
3
votes
2answers
73 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
1
vote
0answers
83 views

Fundamental solution of wave equation in 3D

I want to ask for assistance in verifying the fundamental solution of the wave equation in $\mathbb{R}^{3}$. Here the fundamental solution is given by $$\frac{1}{2\pi}H(t)\delta(t^{2}-|x|^{2})$$which ...
0
votes
2answers
78 views

What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?

For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ...
2
votes
1answer
38 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
2
votes
1answer
92 views

Convergence of a integral - heat Kernel and dirac delta function

Consider $\varphi \in S(R^n)$ (space of rapidly decreasing functions). Consider the heat kernel $$ K_t(x) = \displaystyle\frac{1}{{(4\pi t)}^{n/2}} \displaystyle e^{- \displaystyle\frac{|x|^2}{4t}}, ...
1
vote
2answers
73 views

distributional Laplacean of a function and the dirac delta distribution

Consider $S(R^2)$ the space of rapidly decreasing functions (http://en.wikipedia.org/wiki/Schwartz_space). Consider $F(x) = \displaystyle\frac{1}{2 \pi} \ln|x| , x \in R^2 - \{ 0 \}$. I want to ...
2
votes
1answer
67 views

Continuity condition for distributions in Rudin Functional Analysis

On page 156 of Rudin's Functional Analysis, he gives the following condition for a linear functional over the test functions $D(\Omega)$ to be continuous: A linear functional $\Lambda$ on ...
0
votes
1answer
25 views

For every $A\in \mathcal{L}(C^\infty(\mathbb T^n))$ exists $k_A\in C^\infty(\mathbb T^n\times \mathbb T^n)$ such that..

does anyone know whether it is true that for every $A\in\mathcal{L}(C^\infty(\mathbb T^n))$ there exists $k_A\in C^\infty(\mathbb T^n\times \mathbb T^n)$ such that, $$ Af(x)=\int_{\mathbb T^n} k_A(x, ...
4
votes
2answers
92 views

If $u$ and $v$ have weak derivatives,what about $uv$?

$\Omega$ is a domain in $R^n$, Let $u\in L^1_{\text{loc}}(\Omega)$. If there exists $g_i \in L^1_{\text{loc}}(\Omega)$ such that $$\int_\Omega g_i \phi \, dx=-\int_\Omega u \frac{\partial ...
2
votes
1answer
107 views

$ \frac{\partial^2 T}{\partial x\partial y} = 0 $, then $ T = ? $

Can we characterize all distributions $T \in \mathcal{D}'(\mathbb{R}^2) $ with the following property of distribution derivatives ? $$ \frac{\partial^2 T}{\partial x\partial y} = 0 $$ For functions it ...
2
votes
0answers
71 views

Verifying the fundamental solution of Stokes flow in R^3?

Recall that Stokes flow in $\mathbb{R}^3$ is the solution $(\textbf{u}, p)$ of $$\begin{align} -\nabla p + \mu \Delta \textbf{u} &= \textbf{f} \\ \nabla \cdot \textbf{u} &= 0, \end{align} ...
2
votes
1answer
47 views

A not smooth distribution

Is exist the not smooth distribution which satisfying: $\left ( D_{t}^{2}-D_{x}^{2} \right )u(x,t)=0$ I can't find at least one not smooth distribution like this... Thanks for the help!
3
votes
1answer
45 views

The classification of possible singular supports

I need to find the solutions of $D_{x_1}u=0$ on $\mathbb{R}^{n}$ and to classify the possible singular supports. Any one have an idea how to solve this kind of question? Thanks!
3
votes
2answers
164 views

Differences between $C_c^\infty[0,T]$ and $C_c^\infty(0,T)$

I believe it is true that: If $f \in C_c^\infty(0,T)$, then $f(T)=f(0)=0$. $C_c^\infty(0,T) \subset C_c^\infty[0,T]$ $C^\infty(0,T) \subset C_c^\infty[0,T]$ If $f \in C_c^\infty[0,T]$, it doesn't ...
3
votes
1answer
150 views

PDE with a Dirac Delta function as boundary condition

I would like to have some information how to solve this PDE: $\partial_tu(x,t)=k^2\partial_{xx}u(x,t)$ with the following boundary and initial conditions: ...
2
votes
1answer
72 views

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in ...
8
votes
2answers
167 views

A Differential operator.

What are the fundamental solutions for the operator $$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$ on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$. Here ...
3
votes
1answer
79 views

Differential equation on $\Bbb R$

We have a differential equation on $\Bbb R$ of the form $$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$ where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a ...
6
votes
0answers
229 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
1
vote
1answer
96 views

Solving the equation $\nabla u=f$.

Let $\Omega\subset\mathbb{R}^N$ be a open set, $\mathcal{D}(\Omega)=C_0^\infty(\Omega)$ and $D'(\Omega)$ the set of distributions. Suppose that $f_i\in \mathcal{D}'(\Omega)$ for $i=1,\ldots,N$. Define ...
3
votes
0answers
44 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
2
votes
1answer
44 views

Asymptotic behaviour of solutions to elliptic PDE

Let $u$ be a solution (in the distributional sense) of $$ \Delta u = \delta_r $$ on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$. Let $w$ be a solution of $$ Aw = \delta_r $$ where $A = ...
2
votes
0answers
82 views

Can we say approximation of Sobolev function by a smooth function is inspired from “Weierstrass approximation”? [closed]

As we see that the idea of approximation of a function in Sobolev space by a smooth function looks like an abstraction of Weierstrass approximation theorem..
0
votes
1answer
130 views

Show that $\delta(\xi-x)=\delta(x-\xi)$

How would you show $\delta(\xi-x)=\delta(x-\xi)$ if you know that $$\int _{-\infty}^{\infty}\delta(x)h(x)=h(0)$$ Also how would you then show more generally that if $f(\xi)$ is a monotonic ...
1
vote
1answer
77 views

Find $r(x)$ such that $r(x)L$ is self-adjoint

The differential operator $$L=a(x)\frac{d^2}{dx^2}+b(x)\frac{d}{dx}+c(x)$$ is not self adjoint. How would you find r(x) such that r(x)L is self adjoint. I know that this is self adjoint when $L=L^*$ ...
5
votes
4answers
223 views

delta functions $e^{x}\delta (x)=\delta (x)$

How would you prove that; $$e^{x} \delta (x)= \delta (x)$$ Is it anything to do with the following relationship; $$ \int_{-\infty}^{\infty} g'(x)h(x)\,dx = \int_{-\infty}^{\infty} g(x)h'(x)\,dx.$$ ...
1
vote
0answers
37 views

Elliptic regularity on intervals

Let $I = ]a,b[ \subset \mathbf{R}$ be a bounded interval, and $L = \partial^n + c_{n-1} \partial^{n-1} + \cdots + c_1 \partial + c_0$, where $c_i \in C^\infty(\overline I)$. Suppose $u \in L^2(I)$ is ...
2
votes
0answers
136 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
1
vote
1answer
110 views

Laplacian of the potential function

Given a potential function $$\Gamma(x)=\frac{1}{(n-2)|S^{n-1}||x|^{n-2}}$$ where $n\ge 3$ is an integer, $|S^{n-1}|$ is the volume of the sphere, $x$ is in $n$ dimension. Is it true that $$-\Delta ...
8
votes
2answers
584 views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
2
votes
1answer
313 views

Justification of use of delta functions/rigorous proof of Green's function for Poisson equation

I'm looking at the proofs of Helmholtz's theorem, but I'm having trouble justifying their interchange of integration and differentiation and subsequent treatment of the "integrand" as a delta ...
3
votes
2answers
84 views

No distribution for associated function

Show that there is no distribution $f \in D'(\mathbb{R})$ such that \begin{equation} f(\phi)=\int e^{1/x^2}\phi(x)dx \end{equation} for every $\phi \in C_{0}^{\infty}(\mathbb{R})$ with $supp(\phi) ...
4
votes
1answer
124 views

Generalizing the weak derivative

I am wondering about the weak derivative in time. We say f has a weak derivative f' if $$\int_0^T f\phi' = -\int_0^T f'\phi$$ for all $\phi \in C_0^\infty(0,T)$. This definition uses the $L^2$ inner ...
-1
votes
1answer
200 views

Fundamental solutions of PDEs

I have two questions about solving PDEs. $L$ is an linear differntial operator In the complement of the origin, the equation $LE =\delta$ reduces to $LE = 0$. Why? What can say about solutions of $ ...
-1
votes
1answer
64 views

When Dirac function is in $H^{-m}(R^n)$?

If Dirac function $\delta\in H^{-m}(R^n)$,please give the range of $m$?
1
vote
1answer
218 views

Singular support of distributions

I've read the following definition: Let $u \in \mathcal{D}^\prime(\mathbb{R}^n)$. We say that $y_0 \notin \mathrm{sing} \ \mathrm{supp} \ u$ if there exists $\phi \in ...
2
votes
1answer
138 views

Extension of formula for solution of heat equation on distributions

Consider a heat equation $$ u_{t} = u_{xx}+f(t,x),\; (x,t) \in (0,L)\times(0,T] \\ u(0,x) = 0, \; x \in [0,L] \\ u(t,0) = 0, \; t \in [0,T] \\ u(t,L) = 0, \; t \in [0,T]. $$ If $f(t,x)$ is ...
2
votes
0answers
67 views

Does multiplication commute with taking of fundamental solution (heat equation)

Let $\Phi(t,x)$ be a heat function, $$ \Phi(t,x) = \frac{1}{\sqrt{4 \pi t}} \exp\left(-\frac{x^2}{4t}\right). $$ Then $(\partial_{t} - \partial_{xx})\Phi(t,x) = \delta(t)\delta(x)$. Furthermore, ...