1
vote
1answer
22 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
1
vote
0answers
37 views

An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
3
votes
0answers
48 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
0
votes
1answer
28 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
2
votes
0answers
59 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
2
votes
1answer
49 views

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that ...
0
votes
1answer
13 views

Do 3-D vectors of distributions (specifically vectors containing delta functions) have Helmholtz decompositions?

Define the function $f_i:\mathbb{R}^3\rightarrow\mathbb{R}^3$, $i\in\{1,2,3\}$, by $f_i(\boldsymbol{x})=\delta(\boldsymbol{x-x_0})\boldsymbol{e}_i$ where $\delta$ is the Dirac Delta function and ...
4
votes
1answer
56 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
1
vote
0answers
35 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
1
vote
1answer
53 views

show a PDE has no distribution solution in $\mathbb{R^2}$\{0}

" Consider the following equation in the plane $x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=f(x^2+y^2)$ where $f(t)$ is a $C^\infty$ function of the real variable $t$ such that ...
0
votes
1answer
64 views

Verify that a function is a solution to the 3-dimensional wave equation.

For $n=3$. How to verify that a smooth function $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{*} $$ is a solution to the 3-dimensional wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad ...
1
vote
1answer
191 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where ...
0
votes
0answers
36 views

Smoothness of distributions

I've reached an impasse in reading some texts on distribution theory, as several of them mention smooth distributions, but none of them actually define what it means. Therefore I'd like to know if ...
1
vote
0answers
41 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
103 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
52 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
59 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
38 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
79 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
67 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
59 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
103 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
83 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
370 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
79 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
81 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
104 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
81 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 ...
0
votes
1answer
53 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
126 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
votes
2answers
93 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
68 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
107 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
132 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
78 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
49 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
56 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
186 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
206 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
75 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
177 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
80 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
244 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
102 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
45 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
46 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
86 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
138 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
81 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 $L=L^*$ ...