Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Singularity removal theorem for $L^2-$bounded harmonic $2-$forms whose singular set has Hausdorff co-dimension at least 4.

I'm trying to study G. Tian's paper Gauge theory and calibrated geometry I and I would greatly appreciate any help to understand the following argument in the proof of Corollary 2.3.2: Since ...
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14 views

Closedness of the range of differential operator first order

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j ...
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1answer
21 views

Control of $L^{\infty}$ Norm of 3d Heat Equation Solution for $L^{3}$ Initial Data

Let $w_{t}$ denote the 3-dimensional heat kernel $$w_{t}(x)=(4\pi t)^{-3/2}e^{-\left|y\right|^{2}/(4t)},\qquad y\in\mathbb{R}^{3}, \ t > 0$$ Suppose $f\in L^{3}(\mathbb{R}^{3})$, and let ...
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20 views

Why L belongs to the dual space $H^{-1}$

I'm studying pde using Evans book. In chapter 6 he introduces second order partial differential operators for example : $L= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}$ I can't understand why $L \in ...
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11 views

2D Poisson equation and Bessel Functions

If I were to solve a inhomogeneous 2D Laplace's equation (in polar coordinates), of the form: $$\nabla^2 f= J_2(r)$$ How would I use the Hankel or the Mellin transform to find a solution? I couldn't ...
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31 views

{$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$

I have a question that it maybe simple but I can not understand why we have : {$\mathbb u$ $\in W^{2,2}(\Omega)$ , such that $u=0$ , $ \Delta u=0 $ on $\partial \Omega $} $\subseteq$ ...
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21 views

Help with method of characteristics question.

I'm trying to solve the partial differential equation, $$ \frac{\partial F}{\partial t} = (z-1)\left(kF - d \frac{\partial F}{\partial z}\right) $$ where $k$ and $d$ are constants greater then $0$, ...
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1answer
22 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...
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27 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
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18 views

Evans PDE Chapt 5 problem 4, existenc of smooth functions form a partition of unity

I have to be honest that I am very lost on the same kind of problem about proving existence of smooth function. I have not done much topology, hence unfamiliar with the "covering" business. In the ...
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35 views

Clarification on the validity of separable solutions to PDEs

So we know that various famous PDEs have solutions that can be found by assuming the function is a product of functions each in one variable, e.g. 'Let $V(a,b,c)=A(a)B(b)C(c)$' etc. What I'd like to ...
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1answer
12 views

Calculating Laplacian after substituting polar coordinates in derivation of fundamental solution to Laplace's Equation

I'm following the derivation of the fundamental solution to Laplace's equation in section 2.2.1 of Evans's PDE book. It's the standard approach. We assume a radially symmetric solution $v(r)$ and do ...
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0answers
11 views

Nonlinear-Variation of Helmholtz Equation

I was wondering on the solution of the equation $$\nabla^2P(\vec r)=v(\vec r)P(\vec r)^2\phantom{.......}(1)$$ Or more simply, if there exists a coordinate system where: $$\nabla^2P(\vec r)=P^2(\vec ...
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1answer
53 views

Analytical solution to complex Heat Equation with Neumann boundary conditions and lateral heat loss

I have solved a PDE in this from numerically on Mathematica, but does anyone know if there is a way to solve the following PDE analytically, an analytical solution would really help me. This is an ...
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20 views

Applying the BDF2 formula to the KdV equation

I'm trying to numerically solve the KdV equation $$u_{t} + u_{xxx} = 0$$ with IC $$u(x,0) = 2 \text{sech}^{2}(x)$$ on Julia. Here, I want to discretise the spatial term, $u_{xxx}$, using the BDF2 ...
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18 views

A Sub-Laplacian equal to $\displaystyle\sum_{k=1}^n \partial_{x_k}^2$

Let $\mathbb G=(\mathbb R^n, \star)$ be Homogeneous Lie group (or a stratified Lie group) and let $L$ be a subLaplacian on $\mathbb G,$ defined by $$ L_{\mathbb G}=-\sum_{1}^{n}X_{i}^{2} $$ ...
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40 views

A system of partial differential equations

I have 6 partial differential equations that in the first look they don't seem very difficult, but all my efforts for solving them were unsuccessful. $$\frac{\partial f(x,y,z)}{\partial ...
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1answer
29 views

What is the difference between single and double layer potential

I want to know the difference between single-layer and double layer potentials. Is there a link between the choice of single/double layer potential and the boundary condition of a PDE or an ...
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0answers
12 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
2
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1answer
47 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
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0answers
13 views

Relationship between Faedo-Galerkin Method and Semigroup Method

I have multiple questions relating to Galerkin Method and the Semigroup Method of proving existence of solutions to PDEs. In the Galerkin Method, we decide on a function space, find eigenfunctions ...
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26 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
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1answer
24 views

Stepping backwards with Forward Euler?

Let us say I want to use Forward Euler scheme to solve the heat equation $$ \frac{\partial u}{\partial t} = -\frac{\partial^2 u}{\partial x^2} $$ in the domain $t \in (0, 1)$, $x \in (0,1)$ but ...
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1answer
26 views

principle maximum for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset R^n$, with Dirichlet boundary conditions. For all $f\in L^2(\Omega),$ we denote by $S(t)f$ the solution of the equation $$ ...
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15 views

Noncharacteristic planes for nonzero PDO [on hold]

How to prove that for nonzero partial differential operator of degreem most planes are noncharacteristic?
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1answer
21 views

Operator equation $Au = f$ for $-\Delta u(x)=f(x)$

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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1answer
41 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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1answer
48 views

Intuition behind eigenfunctions of the Laplacian operator

I'm reading about the notion of spectral dimension which is a measure of how particles diffuse in some space at different scales. An important aspect of spectral dimension is the ...
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23 views

Adi-method for Diffusion-reaction equation in 2d

i'm trying to solve this pde using an adi-method (alternating-direction-implicit). $\frac{d f}{d t}=D\nabla^2_{x,y} f+Q(x,y)f+C$ After discretizing, the equation looks like this. Implicit in ...
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16 views

Non-linear hyperbolic systems

This question is about a naive approach to non-linear hyperbolic systems, thinking in the context of elasticity. To set up the problem suppose $\Omega\subset \mathbb{R}^n$ is open and bounded. ...
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19 views

Regularity of compactly supported solutions to the divergence equation: $\nabla\cdot \mathbf{v}=g$.

I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An ...
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1answer
24 views

Does smoothness imply boundedness? Evans PDE chapter 2 Problem 18

In problem 18, enter link description here 1) I am having difficulty in extracting information in deciding the bounded for $g$ and $h$. In particular, to conclude $g$, $Dg$, $h$ $Dh$ are bounded by ...
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1answer
34 views

How to evaluate this combination of sums and integrals?

I am reading a book on PDEs, and I am near the beginning where the author is talking about the heat equation and, specifically, solving the non-homogenous equation $u_t={\alpha}^2u_{xx}+f(x,t).$ The ...
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1answer
25 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
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1answer
16 views

about convection term in the NSE

Let $u=(u_1(x, y), u_2(x, y))$ be two dimensional vector field and consider the convection term $(u\cdot\nabla) u$ in the NSE. In some books it is usually written in this form, but in some other books ...
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1answer
56 views

Operator theory curiosity

I'm not an expert in operator theory... but i was wandering if there's some practical applications. For example (the first one i came up with) compared to normal calculus techniques that usually the ...
2
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1answer
24 views

First Eigenfunction of Simple Equation

Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(\frac{\pi n}{a}x)$ ...
0
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1answer
53 views

PDE on a Damped Wave Equation

What are the eigenvalues and eigenfunctions of $$ X''+X'-\sigma X=0 \\ \text{ with boundary condition } X(0)=X(l)=0$$ I know that for $X''-\sigma X=0 $, the eigenvalues would be $ -\left ( \frac{n ...
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34 views

What math is needed for the inverse quantum problem?

What sort of mathematical background/familiarity is necessary and/or useful in tackling the inverse quantum problem? As an applied math major with a physics minor, I'm looking at different senior ...
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0answers
16 views

reversibility scalar conservation law

I am reading here and there (see for instance Denis Serre systems of conservation laws 1- p.36), the following for which I can't spot the mistake I am making that prevents me from arriving to the same ...
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0answers
27 views

An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = ...
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3answers
25 views

solution of wave equations in odd dimension Evans PDE

Here I am looking at the proof of theorem 2 below Here I have the following difficulties: 1) In the last two lines, the exponent changes from $\frac{n-1}{2}$ to $\frac{n-3}{2}$, why? Could anyone ...
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1answer
30 views

show that $uv\in W^{1,r}(\Omega)$

Let $\Omega\subset\mathbb{R}^{n}$ is bounded,with $\partial\Omega\in C^{1},p,q\geq 1$,$u\in W^{1,p}(\Omega),v\in W^{1,q}(\Omega)$,show that $ uv\in W^{1,r}(\Omega)$.here ...
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1answer
33 views

Prove $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$ [duplicate]

Prove that for $n>1$,the non-bounded function $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$,Here $\Omega=B(0,1)\subset \mathbb{R}^{n}$ I think we have to prove that $$ ...
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3answers
57 views

prove that $u$ is equal a.e. to an absolutely continuous function

Prove that if $n=1$ and $u\in W^{1,p}(0,1) $ for some $1\leq p<\infty$, then $u$ is equal a.e. to an absolutely continuous function,and $u'$ (which exists a.e.) belongs to $L^{p}(0,1)$. My ...
2
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0answers
35 views

Method of characteristics for a parabolic PDE

I am trying to solve a second order parabolic PDE using the method of characteristics. The PDE has the following form: $$\alpha\frac{\partial^2u}{\partial x^2}-\gamma\frac{\partial u}{\partial ...
2
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1answer
19 views

Convolution with standard mollifier

Let $\Omega \subset \mathbb{R}$ open and $f \in L^p(\Omega).$ Now, we define $$\eta(x):=\chi_{[-1,1]}(x) e^{\frac{-1}{1-x^2}}.$$ Then we define $$\eta_h(x):=\frac{1}{h} \eta( \frac{x}{h}).$$ This ...
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27 views

Interpretation of a certain transform

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ ...
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0answers
11 views

How to construct a fundamental solutions of a PDE from well-posedness?

A fundamental solution of a linear operator $P$ on a manifold $M$ is a distribution $G$ such that: $$P(G)=\delta(x-y)$$ In formal terms this is stated as given a test function $\phi$ then: ...
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1answer
16 views

Perturbation of PDE and Green's function

If the Green's function of a second order differential operator $L$ is $G^{(0)}$, then if I add a small perturbation $\delta L$, a Green's function $G$ for the operator: $(L+ \delta L)$ should be: ...