1
vote
0answers
31 views

Dirichlet Eigenvalues of Laplace-Beltrami operator in Hyperbolic space

Consider the hyperbolic half-plane $\mathbb{H}=\lbrace (x,y)\in\mathbb{R}^2: y>0 \rbrace$ with standard Riemannian metric. The Laplace-Beltrami Operator can be written as ...
1
vote
1answer
21 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 ...
4
votes
3answers
115 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
2
votes
1answer
67 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
2
votes
1answer
30 views

$1/r^2\int_{\mathbb{S}_r}u-u(x)$ converging to $\Delta u(x)$?a

When reading some papers on PDEs, the following shows up several times: For a $C^{\infty}$ function $u$, $\frac{\int_{\mathbb{S}_r(x)}u-u(x)}{r^2}$ converges to $1/2n\Delta u(x)$ uniformly on ...
0
votes
1answer
28 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
1
vote
2answers
57 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
0
votes
1answer
34 views

Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
1
vote
1answer
68 views

Method of Characteristics for a non-linear PDE

I've been trying to work through some of the more difficult questions we've been given in class in regards to the method of characteristics for solving PDEs, but I've come a bit unstuck. I've been ...
0
votes
1answer
24 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
0
votes
0answers
26 views

Di Perna-Lions theory

I'm reading the paper of Di Perna and Lions "Ordinary differential equations, transport theory and Sobolev spaces". I'm not understanding the proof of corollary II.2; in particular I don't understand ...
0
votes
2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
1
vote
1answer
43 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
0
votes
0answers
26 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
1
vote
1answer
34 views

Riesz measure associated with a subharmonic function

In page 101, corollary 4.3.3., from Armitage and Gardiner's book on potential theory, the authors prove that any subharmonic function, can be identified with a positive measure (Riesz measure). In ...
0
votes
1answer
27 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
46 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
31 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
33 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
30 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
33 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
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
40 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
50 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
16 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
1answer
104 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
40 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
27 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
37 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
81 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
45 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
29 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
111 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
94 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
74 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
60 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
49 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
29 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
74 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
135 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
51 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
57 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
59 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 ...