0
votes
1answer
15 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
2
votes
1answer
44 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
1
vote
0answers
14 views

existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
0
votes
0answers
15 views

intuitive fact of a class of functions defined in $R^n$

I am reading an article and i have the following situation: Let $u: R^n \rightarrow R$ a continuous function in $R^n$. Supoose that u is nonnegative and that for all $t \geq 0$ the set $L_t = \{ x ...
1
vote
0answers
25 views

Di Perna-Lions theory for transport equation

Does someone know if some notes on the topic mentioned in the title are available online? I'm reading the paper "Ordinary differential equations, transport theory and Sobolev spaces" by Di Perna and ...
1
vote
1answer
25 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
0
votes
0answers
28 views

Basic exercise of real analysis and of p harmonic functions

I am studying the following definition of an article: Definition (blow up): Let $u$ a function (assuming real values) in the open ball $B(x_0,1)$. For $r>0$ define the function $u_r(x)$ in $B(0, ...
0
votes
1answer
23 views

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
1
vote
1answer
32 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
0
votes
1answer
43 views

Definition of weak divergence [closed]

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
0
votes
1answer
9 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
1
vote
0answers
37 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
0
votes
1answer
49 views

Green's representation on a compact domain

This is from page 17-18 of Trudinger and Gilbarg Let $\Omega$ be a domain for which the divergence theorem holds. Let $\Gamma(x-y)$ be the normalised fundamental solution of the Laplace's equation, ...
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
60 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 $\beta \in \mathbb R$. ...
1
vote
1answer
39 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
1
vote
1answer
25 views

A priori estimates for functions in $C_0^\infty (\overline{\Omega})$.

Let $u\in C_0^\infty (\overline{\Omega})$, where $\Omega\subset \mathbb{R}^N$ is a bounded domain. Fix some $a\in \Omega$ and choose $r>0$ such that $\overline{\Omega}\subset B(a,r)$. Define ...
1
vote
0answers
35 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
3
votes
0answers
73 views

Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad ...
1
vote
1answer
44 views

Energy of a solution of the wave equation.

Let $f\in\operatorname{C}^2(\mathbb{R})$ and $g\in\operatorname{C}^1(\mathbb{R})$ be function whose support are compact. By considering a solution $u$ of the problem $$ \begin{cases} u_{tt}(x,t) - ...
0
votes
1answer
28 views

The spherical mean of $h(x,y) = x$.

The spherical mean of a function $h : \mathbb R^2 \to \mathbb R$ is given by $$ \frac{1}{2\pi r} \int_0^{2\pi} h(x + r \cos(\theta), y + r \sin(\theta)) d \theta $$ Now I want to compute the ...
10
votes
1answer
108 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
5
votes
0answers
88 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
1
vote
2answers
66 views

Inverse Laplace operator $\Delta^{-1}$ and Sobolev spaces

I'm looking for some regularity results for the inverse Laplace operator. More precisely - we're set in $\mathbb{R}^3$ and we are looking at the operator $$ \Delta^{-1}f = \frac{x}{|x|^3} \ast f$$ I'd ...
2
votes
3answers
57 views

2D Heat Equation with special initial condition

I want to solve the 2 dimensional heat equation on a square $\Omega = \{ (x,y) : 0 < x < \pi, 0 < y < 2\pi \}$ with the Fourier Method \begin{align*} \partial_t u - \Delta u & = 0 ...
1
vote
1answer
48 views

proof of coarea formula for n dimensional hypersurface in $R^n$

$f:R^n \rightarrow R$ be continuous and summable. please give the proof for these formulas $\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr$ $\frac{d}{dr}\int_{ ...
1
vote
0answers
27 views

Solving $u_{\xi\eta} = 0$ and differentiability conditions on solutions

After transformation someone often encounters the PDE $$ u_{\xi\eta} = 0 $$ but I am quite confused about the differentiability conditions of its solution (for example in this post I read different ...
1
vote
2answers
73 views

On the abstract bootstrap principle in the book “Nonlinear Dispersive Equations” by Terence Tao

In Terence Tao's book "Nonlinear Dispersive Equations", he gives the following "Abstract bootstrap priciple": "Let $I$ be a time interval, and for each $t \in I$ suppose we have two statements, a ...
2
votes
3answers
132 views

Computing the spherical mean and showing it satisfies PDE

Compute the spherical mean of the function $h : \mathbb R^3 \to \mathbb R$ with $$ h(x,y,z) = x $$ and show that it satisfies the differential equation $$ u_{rr} + \frac{2}{r} u_r = u_{xx} + u_{yy} ...
0
votes
0answers
38 views

What happened to Otelbayev's proof on Navier-Stokes existence and smoothness? [duplicate]

I haven't heard any since the last comment on Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? So what has gone on then? Sorry for asking a too soft question here. ...
0
votes
1answer
50 views

negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
1
vote
0answers
13 views

Continuity of the solution to a matrix PDE (mapping of a parameter to solution)

I'm considering the following PDE in $\Phi$: $\frac{\partial \Phi(t,s)}{\partial t}$ + $sR\frac{\partial \Phi(t,s)}{\partial s}$ + $\frac{1}{2} s^2 M \frac{\partial^2 \Phi(t,s)}{\partial s^2}$ + ...
0
votes
1answer
69 views

An equality from Fritz John's paper

Prove: $\frac{\rho}{(4\pi)^2}\int_{|\xi|=1}d\omega_\xi\int_{|\eta|=1}f(x^0+r\xi+\rho\eta)d\omega_\eta=\int_{|r-\rho|}^{r+\rho}\frac{\lambda}{8\pi ...
1
vote
1answer
56 views

Convergence weak* in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ implies convergence weak* a.e in $L^2(\mathbb{T}^2)$?

Suppose $x_n\rightharpoonup x$ in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ with the weak* topology, in other words, $\forall f\in L^{1}([0,T];L^2(\mathbb{T}^2))$ we have $$\lim_{n \to\infty} \int _0 ^T ...
0
votes
1answer
43 views

Tranforming boundary value problem (heat equation) to one with homogenous boundary condition

Let $\Omega \subseteq \mathbb R^n$ be a bounded region with smooth boundary, let $u$ be two times continously differentiable in $\Omega \times (0,\infty)$ and in $\overline \Omega \times [0,\infty)$ ...
1
vote
0answers
13 views

Finding the continuity of the mapping of a solution to a PDE to its partial derivative

Here is a modified version of the Black-Scholes PDE: $\frac{\partial \phi(t,S,i)}{\partial t}$ + $r_iS\frac{\partial \phi(t,S,i)}{\partial S}$ + $\frac{1}{2} \sigma^2_i S^2 \frac{\partial^2 ...
-1
votes
1answer
25 views

Solving two equations for $p$ and $q$

How we obtain from the equations $z^2-pqxy=0$ and $z-a(xp+yq)=0$ that $p=\frac{z}{cx}$ and $q=\frac{cz}{y}$, where $a(c+\frac{1}{c})=1$?
2
votes
0answers
26 views

Missing Step in a Paper of Struwe.

In this paper on Page 4, in the last line of the proof, the author asserts that if a radial function $u:\mathbb{R_t}\times \mathbb{R}^2\to \mathbb{R}$, smooth outside the origin $(0_t,0_x)$, admits ...
0
votes
1answer
40 views

Differentiation Formula for Moving Regions.

I've run into a few calculations in a series of textbooks/papers that require differentiating an integral with a changing region. In particular, I'd like to know if $f(x,t):\mathbb{R}^d\times ...
6
votes
1answer
72 views

counterexample for ill posedness of the laplace equation

Consider the wave equation with initial data: $$u_{tt}(t,x) + u_{xx}(t,x) = 0$$ $$u(0,x) = u_0(x)$$ $$u_t(0,x) = u_1(x)$$ Hadamard showed that this problem is ill-posed: there exist large solutions ...
3
votes
1answer
33 views

Smooth function composed with sobolev function vanishes at 0

Let $\Omega$ be a bounded domain with sufficiently smooth boundary. Let $u \in W^{1, 2}_{0}(\Omega)$ and $F \in C^{\infty}(\mathbb{R} \rightarrow \mathbb{R})$ such that $F(u(x)) = 0$ for almost every ...
0
votes
1answer
60 views

Minimizing cost function (Eikonal)

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $S \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=S$ and ...
0
votes
0answers
29 views

Laplace's equation boundary conditions

I am supposed to determine a solution from the following boundary conditions in a rectangular area $V$. $f(0,y)=f(x,a) = f(x,0)=0$ and $f(b,y) = C\ sin(\frac{\pi}{a}y)$, for $a,b \in \mathbb{R}$. I ...
0
votes
1answer
60 views

Eikonal Equation [duplicate]

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $A \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=A$ and ...
1
vote
0answers
20 views

Consequence of elliptic estimates up to the boundary

Consider the half space $\Omega=\{x=(x_1,\ldots,x_N)\in\mathbb{R}^N:x_N>0\}$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ a positive bounded solution of $$ \begin{eqnarray*} \Delta ...
0
votes
0answers
17 views

Making sense of Duhamel.

If I'm solving, say, cubic NLS. I write out the contraction mapping, inspired by Duhamel, and inevitably am required to write down something of the form ...
1
vote
1answer
23 views

Inequality in $L^2$ based norms

Recently I came across the following inequality. Let $f:\,\mathbb{R}^n\rightarrow\mathbb{R}$ be smooth, then the inequality $$\left\lVert f\right\lVert_{\dot H^2(\mathbb{R}^n)}\leq\left\lVert\Delta ...
1
vote
1answer
31 views

the differentiability from an oscillation estimate

If we have an oscillation estimate of $u\in W^{2,n} (B^{+}) \cap C^0(\bar B^{+})$, and $u=0$ on $T = B \cap \partial R^{n}_{+}$, that $$osc_{B^{+}} \frac{u}{x_{n}} \leq C.$$ Then how comes that $u$ is ...
0
votes
1answer
34 views

Maximum principle type problem

Suppose $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ is sufficiently smooth and is such that $(1/2) u_{xx} + u_{xy} + 2u_{yy} = 0$ in a ball $B$ centered at the origin. Must $u$ attain a maximum inside ...
1
vote
2answers
52 views

Why does Boundary $H^2$ regularity fail for trace non-zero functions?

In Evans, Section 6.3, Theorem 4; we know that if $\Omega$ is a bounded region in $\mathbb{R}^n$ with smooth boundary (say), and $u\in H_0^1(\Omega)\cap H^2(\Omega)$ then we have the bound ...