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

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6
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1answer
204 views

Capacity theory beginner resources

I'm currently studying a book on shape optimization: Variation et optimisation de formes: Une analyse géométrique By Antoine Henrot, Michel Pierre. The book introduces at some point capacity, and uses ...
6
votes
2answers
2k views

Finding integrating factor when IF will be a function of x and y

I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y. I'm on this problem, $$ ( y - xy^2 ) ...
5
votes
1answer
108 views

How does the Hahn-Banach theorem implies the existence of weak solution?

I came across the following question when I read chapter 17 of Hormander's book "Tha Analysis of Linear Partial Differential Operators", and the theorem is Let $a_{jk}(x)$ be Lipschitz continuous in ...
5
votes
2answers
99 views

Looking for a first order perturbation of the Laplacian having 0 in its spectrum

I would like to find an explicit example of a linear elliptic operator having the following form: $$Lu=-\Delta u +b(x)\cdot \nabla u, $$ where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that ...
5
votes
1answer
223 views

How should a numerical solver treat conserved quantities?

Since the good old Noether Theorem ultimately states that any physical system will exhibit conserved quantities, how should they be treated best in numerical solvers? On the one hand, one can observe ...
5
votes
2answers
294 views

How to use FEM to solve a PDE

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. ...
5
votes
1answer
419 views

Eigenvalues For the Laplacian Operator

How do I show that the asymptotic speed of the eigenvalues $\lambda$ of the Laplacian Operator is $O(m^{2/n})$ where $m$ is the index of the eigenvalues and $n$ is the dimension of the space?
4
votes
1answer
61 views

How find a solution to this PDE $\frac{xf'_{x}}{f'_{y}}+\frac{yf'_{y}}{f'_{x}}+x+y=C$

let $C$ is give the constant ,if the function $f(x,y)$ such $$\dfrac{xf'_{x}}{f'_{y}}+\dfrac{yf'_{y}}{f'_{x}}+x+y=C$$ Find the all $f(x,y)$ I found this problem one solution: ...
4
votes
1answer
146 views

Some Scaling Estimate for Heat Kernel

NOTE. I have rewritten the question to summarize my current progress on this question. The bounty is for completing what I have done so far, or by offering a more elegant solution probably based on ...
4
votes
1answer
140 views

Need help understanding this proof of regularity of traveling wave solutions to the Gross-Pitaevskii equation

These are actually 4 question about a proof given in this paper. Any hint to solutions for any of these questions would be much appreciated! Lemma 1. Assume $v$ is a solution to the equation ...
4
votes
0answers
118 views

Is there any theorem that tells us how many ICs or BCs are needed for getting the determine solution of a PDE or a set of PDEs?

It's a shame that, though I've taken the "Equations of Mathematical Physics" class for one semester and solved numbers of PDEs with Mathematica, I'm still unclear about how many initial ...
4
votes
1answer
527 views

The extension of smooth function

If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...
3
votes
0answers
68 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial ...
3
votes
1answer
472 views

understanding of the classical definition of Green's function

I learn the classical definition of Green's function from Hunter's Applied Analysis. Consider the second-order ordinary differential operators $A$ of the form $$Au=au''+bu'+cu,$$ where $a,b$ and $c$ ...
2
votes
1answer
61 views

A question about a Sobolev space trace inequality (don't understand why it is true)

Let $\Omega$ be an open set with boundary $\partial\Omega$. Let $u \in H^1(\Omega)$. There exists a $\lambda \in \mathbb{R}$ such that $$\int_\Omega |\nabla u |^2 + ...
2
votes
1answer
257 views

Derivation of weak form for variational problem

My question is about understanding the derivation of the weak form of a variational problem (to be used for the solution via the finite element method). The problem is as follows (it is an image ...
2
votes
1answer
115 views

Solve the given Cauchy problem on the bounded interval

$$u_{tt}-16u_{xx}=0, \quad 0<x<3, \quad 0 < t < \infty,$$ $$u(x,0)=x(3-x), \quad u_t(x,0)=\cos(\pi x), \quad 0<x<3,$$ $$u(0,t)=u(3,t)=t, \quad 0 < t < \infty.$$ Determine ...
10
votes
1answer
192 views

A type of local minimum (2)

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is a Lipschitz graph. $S \subset \partial \Omega$ is measurable and $H^{n-1}(S)>0$. ...
8
votes
2answers
165 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 ...
7
votes
1answer
264 views

Do the two limits coincide?

Let $a$ be a non negative (positive almost everywhere) weight in $L_{loc}^1(\Omega)$, $\Omega\subseteq\mathbb{R}^n$ is open. For $\varphi\in C_c^{\infty}(\Omega)$ define $$ ...
5
votes
1answer
138 views

Spherical means (Kirchoff's formula) for variable speed wave equation

Suppose $$ \begin{cases}u_{tt} - \Delta u = 0 \quad \textrm{ in } \mathbb R \times \mathbb R^n \\ u(0,x) = f(x); \quad u_t(0,x)= g(x). \end{cases} $$ then, depending on the dimension $n$, we have a ...
5
votes
4answers
202 views

Why separation of variables works in PDEs?

The method of separation of variables is used in many occasions in the upper level physics courses such as QM and EM. But when it is used there is no clear reason why using it is permitted it except ...
5
votes
1answer
66 views

Need source for elliptic regularity on unbounded domains

I need a source that provides a $W^{2,2}$-regularity result for linear elliptic systems on unbounded domains. To be more specific, I study the equation $$ -\Delta w - ic \partial_1 w + \left( ...
5
votes
1answer
273 views

A PDE exercise from Gilbarg Trudinger. problem 2.2

Prove that if $\Delta u=0$ in $\Omega\subset \mathbb{R}^n$ and $u=\partial u/\partial\nu=0$ on an open smooth portion of $\partial \Omega$, then $u$ is identically zero. I found a proof here, but I'm ...
5
votes
1answer
170 views

Invertibility of laplacian operator

Let $\Omega\in\mathbb{R}^n$ be a bounded open set with smooth boundary. How to prove the invertibility of $$- \triangle:H^2_0(\Omega) \to L²(\Omega) $$ The injectivity is easy. But how to prove ...
5
votes
1answer
297 views

Finding the general solution of a quasilinear PDE

This is a homework that I'm having a bit of trouble with: Find a general solution of: $(x^2+3y^2+3u^2)u_x-2xyuu_y+2xu=0~.$ Of course this should be done using the method of characteristics but I'm ...
5
votes
1answer
203 views

Claims in Pinchover's textbook's proof of existence and uniqueness theorem for first order PDEs

The reference here is Pinchover & Rubinstein's An introduction to partial differential equations, pages 36-37. It's about the existence and uniqueness of a solution to the equation $a(x,y,u)u_x + ...
4
votes
0answers
120 views

Show that the following $u\in L^{\infty}\cap H^1(B)$ is a weak solution to the given system.

Let $B=B_{\exp(-2)}\subset\mathbb{R}^2$. I would like to show that a weak solution to the following system (in $B$): \begin{align*} \triangle u_1&=-2|Du|^2(u_1+u_2)/(1+|u|^2)\\ \triangle ...
4
votes
2answers
168 views

Different types of domains $\Omega \subset \mathbb{R}^n$ in PDEs

In PDEs I often read things like: Let $\Omega$ be a bounded Lipschitz or $C^1$ or $C^2$ or $C^\infty$ domain But I have no clue what this means in real life. I understand ...
4
votes
1answer
146 views

Transient diffusion with compact support throughout (not just initially)

[Pardon my lack of rigor; I am an engineer by training. Also, for convenience, allow me to make this question as concrete as possible.] Assume the simplest linear diffusion equation: $\alpha ...
4
votes
1answer
308 views

Laplace equation Dirichlet problem on punctured unit ball.

Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem \begin{align} \Delta u &= 0 \\ u(0) &= 1 \\ u &= 0 ~~~\text{if} ~~|x|=1 \end{align} By considering ...
4
votes
2answers
345 views

Inverse Fourier transform of a hyperbolic cosine

This problem arises from trying to solve, by Fourier transform, the Cauchy problem $$\begin{cases} u_{tt}-u_{xxxx}=0 &x\in\mathbb{R},\, t\geq 0\\ \begin{cases} u(0,x)=f(x)\\ u_t(0,x)=0 ...
4
votes
2answers
246 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
4
votes
1answer
236 views

A type of local minimum

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ ...
4
votes
2answers
454 views

Non-uniqueness of solutions of the heat equation

For the heat equation $(\partial_t-\partial_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the ...
4
votes
0answers
165 views

Asymptotic behaviour of a two-dimensional recurrence relation

This problem comes out of a research in models of firm growth. The model is simple: A firm has two parameters which are its size (number of employees) and job vacancies. A firm of size $n$ will ...
4
votes
1answer
630 views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically in page 6,equation 8 , not equation 9... There ...
3
votes
1answer
62 views

Solving Wave Equation with Initial Values

I am trying to solve the wave equation: $u_{tt}$ = $u_{xx}$ With initial values: $u(x,0) =\begin{cases} x^3 - x, &\text{for }|x|\le 1,\ \\0, &\text{for }|x|\ge1\end{cases}$ $u_t(x,0) ...
3
votes
1answer
83 views

Show that a Bilinear form is Coercive

I'm reading through Brezis' book on functional analysis, Sobolev spaces and PDE, and I'm having trouble showing that the Bilinear form: $a(u,v) = \int_{0}^{2}u'v'dx+\left(\int_{0}^{1}u dx\right) ...
3
votes
2answers
119 views

about weak derivative of Bochner integrable function

Hi I am studying the following theorem ( the theorem can be found in the classical Evans PDE book in the apendix 5.9 ) Theorem : Suppose $ u \in L^{2}(0 , T ; H^{1}_{0}(U))$ with $u^{'} \in L^{2}(0 ...
3
votes
2answers
177 views

What is the *standard duality argument?

What is the standard duality argument? I saw this foor exemplo in the following statement. The case $p < 2$ follows from the standard duality argument. To prove Theorem: [Calderón Zigmund] If ...
2
votes
1answer
86 views

$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required

Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where ...
2
votes
2answers
51 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
2
votes
1answer
165 views

How to show the curves are conics.

Solve the equation $$\frac{dx}{x^2+b^2} =\frac{dy}{xy-bz}=\frac{dz}{xz+by}$$ How to show the curves are conics. First of all , I need to find its integral curves. I tried to solve. But I get ...
2
votes
2answers
169 views

Solve this Dirichlet problem

Show that the Dirichlet problem $$ \left\{ \begin{array}{l} u_{xx}+ u_{yy}=u^3 \ \text{in} \ x^2+y^2 \lt 1 \\ u=0 \ \text{on} \ x^2+y^2 = 1 \end{array} \right.$$ where $u=u(x,y)$, has only the ...
2
votes
1answer
71 views

Equivalence of norms in $W^{s,p}(\partial\Omega)$

Let $\Omega\subset\mathbb{R}$ be a bounded domain and $p\in (1,\infty)$. Take $s\in (0,1)$ and define $$W^{s,p}(\Omega)=\left\{u\in L^p(\Omega):\ ...
2
votes
1answer
88 views

Simplicity and isolation of the first eigenvalue associated with some differential operators

Consider the operator $\Delta$ or more generally a second order differential operator $L$, in which the principal part is symmetric and positive definite. It can be proved (see here page 336) that the ...
2
votes
1answer
90 views

Laplacian Boundary Value Problem

Given a domain $ M \subset \mathbb{R}^2 $ and a function $ f : \partial M \rightarrow \mathbb{R} $, let $ \omega $ solve the boundary value problem: $$ \Delta \omega = 0 \text{ in } M \\ \omega = f ...
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 ...
2
votes
0answers
92 views

Best method to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$, $$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...