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

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2
votes
1answer
146 views

Can I use inverse inequality for an infinite space, $H^1$?

Let $u \in H^1(\Omega)$ and $Q_0 u$ is a $L^2$ projection of $u$ to a polynomial finite space $P_k(T)$ where $T \in \mathcal{T}_h$ is a finite element and $\mathcal{T}_h$ is a set of all the elements. ...
1
vote
1answer
49 views

How to show this equality?

I need some help for showing the following equality $$\displaystyle \left(\sum_{i=1}^nx_i\right)^m=\sum_{|\alpha|=m}\frac{m!}{\alpha!}x^\alpha$$ for all $x\in\mathbb R^n$. Here $\alpha\in\mathbb N^n$ ...
4
votes
0answers
78 views

Restriction on exponent; Weak Harnack inequality for strong solutions

The weak Harnack inequality for strong solutions goes as follows (Taking $Lu = a^{ij}(x)D_{ij}u + b^i(x)D_iu+c(x)u$ to be elliptic) Let $u\in W^{2,n}(\Omega)$ satisfy $Lu\leq f$ in $\Omega$ for ...
3
votes
1answer
73 views

Construct harmonic function on noncompact manifold

$M$ is a non-compact Riemannian manifold, $p \in M$. Consider Dirichlet problems: $\Delta u = 0$ in ${B_p}\left( i \right)$ ($i = 1,2, \dots $), $u{|_{\partial {B_p}\left( i \right)}} = {f_i}$, ${f_i} ...
3
votes
1answer
36 views

Density of a subspace in $\mathcal{D}(0,T;V)$ under $H^1$ norm

Let $V$ be Hilbert. Let $\mathcal{D}((0,T);V)$ be space of infinite differentiable functions with values in $V$ with compact support. Are functions of the form $$\sum_j \psi_n(t)w_n$$ where $\psi_n ...
0
votes
1answer
59 views

Isometric embeddings with prescribed second fundamental form

I'm looking for some non-rigidity result for isometric embeddings in euclidean space (codimension 2). For example, any isometric embedding of the round $S^2$ into $\mathbb R^3$ is unique up to rigid ...
2
votes
2answers
782 views

Math Subjects that are prerequisites to Finite Element Method

I want to Learn Finite Element Method. What are the Math Subjects that are prerequisite to FEM? I am a mechanical engineer. FEM was not taught to us in College. :( I am profficient in Engineering ...
4
votes
1answer
80 views

Diffusive/Dispersive character of discretization schemes

I am not sure if this is the correct place to post my question, so please correct me if there is a better site. I recently started applying some discretization schemes such as Upwind, Lax-Wendroff ...
3
votes
2answers
240 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
122 views

Trace for $L^\infty$ functions?

I'm considering the following problem. Let $s\in L^\infty ((0,T)\times K)$ for some compact $K\subset \mathbb{R^n}$ be given. Consider the Steklov average in time of $s$, i.e. for $h>0$ and ...
1
vote
0answers
147 views

A contradiction to $C_c^\infty \subset H^1$ not dense?

Now we know that $C_c^\infty(\Omega) \subset H^1(\Omega)$ is NOT dense. We also know that (eg. from Lions' and Magenes) that if $V \subset H \subset V'$ is Hilbert triple, $$\mathcal{D}(0,T;V) ...
-3
votes
1answer
54 views

How to prove $\mathrm{supp} ~ R(t)\in \{|x|<t\}$

Let $R(t)$ define as $$R(t):=F^{-1} \left(\frac{\sin |\xi|t}{|\xi|} \right),$$ how to show that $$\mathrm{supp} ~ R(t)\in \{|x|<t\},$$ where $F$ is the Fourier transform.
1
vote
1answer
81 views

construction of a smooth function using mollifiers

let $r>0$ and $B(x_0, r) \subset R^n$ . My problem is construct a function $u \in C^{\infty}_{0}(B(x_0, 2r))$ using mollification satisfying $$u = 1 \text{ on } \overline{B(x_0, r)} $$ and $$ ...
2
votes
1answer
270 views

Fisher-KPP equation

Can you help me understand how to derive the Fisher-KPP equation? I.e., how can I figure out where it comes from? It is easy to find the derivations of the diffusion equation and the heat equation, ...
2
votes
3answers
78 views

What does $\frac{\partial}{\partial x}(\frac{\partial f}{\partial u})$ mean when $f(x,t), u=x+ct, v=x-ct $?

I'm trying to transform the wave equation $\frac{\partial^2f}{\partial t^2}-c^2\frac{\partial^2f}{\partial x^2}$ using the substitution: $ \\u=x+ct \\v=x-ct $ and using the chain rule for the ...
2
votes
1answer
94 views

does solution to homogeneous heat equation always attains it's maximum at parabolic boundary?

The following is statement of weak maximum principle for heat equation. $(1)$ What does it mean mean by $C(\bar Q_T)$? I mean I know $||f||_{C^k(\Omega)} = \sum_{n=0}^k \sup|f^{(n)}(x)|_{x \in ...
1
vote
1answer
65 views

showing the solution of heat equation decays with time

For following PDE how can I show $||u(t,*)||_{L^2([0,l])} \le a e^{-bt}$ without solving it, where $a>0$, and $b>0$ are constants. let, $l>0$, $S = (0,\infty)\times (0,l)$ and $u(t,x) \in ...
1
vote
1answer
581 views

Non homogeneous heat equation

How to solve the following PDE? \begin{align*} u_t - u_{xx} = tx & \; ; 0<x<\pi, t>0 \\ u(x,0)=1 &\; ; 0\le x\le \pi\\ u_x(0, t) = u_x(\pi , t)=0&\; ; t> 0 \end{align*} For ...
1
vote
1answer
54 views

Question on proof of deformation lemma

on page no 479 of partial differential equation (Evans) how the condition (iii) of deformation lemma is satisfied.
5
votes
2answers
70 views

Examples where $1 \in W_0^{k,p}\left( U \right)$

$M$ is a Riemannian manifold, $U$ is a domain in $M$. Consider the Sobolev space $W_0^{k,p}\left( U \right)$: the closure of $C_0^\infty \left( U \right)$ (smooth functions with compact support) in ...
2
votes
0answers
60 views

weak derivative and the value of a integral

Let $0 < r < R$ and $p>1$ and consider the function $$u(x) = \displaystyle\frac{\displaystyle\int_{|x|}^{R} t^{-1 }dt}{\displaystyle\int_{r}^{R} t^{-1 }dt},$$ if $r < |x|< R$ , and ...
1
vote
1answer
62 views

question about infimun

Let $\Omega \subset R^n$ a open set and $p >1$. Let $K \subset \Omega$ a compact set. Define $$A:= \{ u \in C^{\infty}_{0} (\Omega) ; \ u \geq 1 \text{ on } K\}$$ $$ B:= \{ u \in C^{\infty}_{0} ...
3
votes
0answers
136 views

How to solve the advection equation with spiral motion

The advection equation is : $$\frac{\partial f(x,y,t)}{\partial t} + \nabla_{(x,y)} \cdot (A f)= 0$$ With initial condition $f(x,y,0) = f_0(x,y)$. If the vector $A$ is constant, ie. $A = ...
2
votes
1answer
36 views

basic integral inequality

Consider $0 < r <R $. I have a function $u \in C_{0}^{\infty}(B(x_0,R))$ such that $u=1$ on $\overline{B(x_0,r)}$ . Consider $y \in \partial B(0,1)$ (fixed) . My book says : $$ 1 \leq ...
5
votes
2answers
525 views

Weighted Poincare Inequality

I'm trying to prove a result I found in a paper, and I think I'm being a bit silly. The paper claims the following: By the Poincare inequality on the unit square $\Omega \subset \mathbb{R}^2$ we have ...
2
votes
1answer
67 views

What does $f \in L^p(I,V) + L^q(I,H)$ mean?

What does $f \in L^p(I,V) + L^q(I,H)$ mean? What does the addition mean?? I see this in PDE theory.
1
vote
1answer
1k views

solving heat equation under Neumann boundary conditions

I am having trouble solving following equation \begin{align*} u_t - u_{xx} = 0 & \; ;0<x<L, t>0 \\ u(0, x)=x &\; ; 0<x<L\\ u_x(t, 0) = u_x(t, L) = 0&\; ; t >0 ...
3
votes
1answer
275 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: ...
1
vote
1answer
119 views

Differential inequalities

Let a function$f \in C^1(\mathbb{R}^2) $ be s. t. $$ \frac {\partial f} {\partial x}(x+2y) +\frac {\partial f} {\partial y}(-2x+y) \ge 0,$$ $$ \frac {\partial f} {\partial x}(-x+y) -\frac ...
3
votes
2answers
277 views

General solution for the system of PDEs from the curl of a vector field equaling another

In my vector calculus class, when we were introduced to the curl operator the professor gave us this example: Is it possible to find a vector field $\mathbf{G}$ such that $$\mathbf{F} = \nabla ...
2
votes
0answers
96 views

Asymptotics of Green's function of laplacian

Let $ \Omega $ be a domain in $ \mathbb{R}^2 $ with a Riemannian metric $ g $, and let $\Delta $ be the laplace-beltrami operator induced by the metric, with dirichlet boundary conditions. For $ x,y ...
3
votes
2answers
51 views

Problem with simple laplacian equation

I would like to solve the following PDE: $$ \partial_x^2 u + \partial_y^2 u = -\frac{2 x^2 (x^2-y^2)}{\left(x^2+y^2\right)^2} $$ The right side comes from $ x^2 \partial_x^2 \log(x^2 +y^2) $. ...
1
vote
1answer
31 views

How to solve for $\Gamma(X,t)$ in $\Gamma_{t,X} = S(X,t) \Gamma_X$?

Given this equation $$\frac{\partial^2 \Gamma}{\partial t \partial X} = S(X,t) \frac{\partial \Gamma}{\partial X},$$ how do you solve for $\Gamma(X,t)$? $S(X,t)$ is unknown and we impose conditions ...
1
vote
1answer
60 views

What is periodic solution to a PDE?

If I have a PDE $$u_t = Au + f$$ with conditions $$u(0,x) = u(T,x)$$ then if it has a solution, why is the solution called periodic? Isn't it only true that $u(0) = u(T)$? It does not follow that ...
2
votes
0answers
76 views

Covering argument

In proving Harnak's inequality (I am referring to this article: "On Harnack’s Theorem for Elliptic Differential Equations"Communications on Pure and Applied Mathematics Volume 14, Issue 3 ), Moser ...
1
vote
0answers
135 views

Take the limit of a function inside a Lebesgue integral.

I've been working on the Fundamental solution of Homogenous Heat Equation and I have problem with the following equality. $$\lim_{\epsilon \to 0} \int_{\mathbb{R}} \frac{1}{2\sqrt{\pi}} ...
0
votes
0answers
52 views

Partial Differential to get a Prob Density Function

Suppose that $u(x,t)$ satisfies the partial differential equation $${\partial u\over \partial t} = \frac 12 {\partial^2u\over \partial x^2}$$ for all $t>0$ and $x\in R$, and also that has ...
1
vote
1answer
33 views

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm?

What is the definition of $H^{-k}(\mathbb{R}^n)$ and its norm? How can I understand the fact $$\|f\|_{H^{-k}(\mathbb{R}^n)}=\|(I-\triangle)^{-k}f\|_{H^{k}(\mathbb{R}^n)}.$$
7
votes
2answers
717 views

Product rule of weak derivatives

I am working on proving the following proposition: If $u,v\in {W^1(\Omega)}$ and $uv,uDv+vDu\in L^1_{\operatorname{loc}}(\Omega)$, then we have the product formula $$D(uv)=uDv+vDu.$$ The definition I ...
3
votes
0answers
95 views

Green's Function Divergence

Given a domain $ \Omega \in \mathbb{R}^2 $, and a PDE of the form $ L = a(x) \partial_x^2 + b(x) \partial_y ^2 $ for $ x \in \Omega $ , the green's function $ G(x,y) : \Omega \times \Omega \rightarrow ...
1
vote
1answer
92 views

Why this functional isn't differentiable?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
5
votes
0answers
125 views

Generating Functions, Recursive Polynomials

At the CMFT international conference in Turkey (2009), the following open problem was given: Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
1
vote
1answer
33 views

Differentiation solving

According to the question , answered by martini , I am failing to evoke my memory that how can we write this $$\sum_i \partial_i^2\rho = \frac{D\rho^2 - \sum_i x_i^2}{\rho^3} = \frac{D-1}{\rho}$$
1
vote
2answers
116 views

Show a PDE satisfies a ODE

Suppose that $u(x,t)$ satisfies the partial differential equation $${du\over dt} = \frac 12 {d^2u\over dx^2}$$ for all $t>0$ and $x\in R$, and also that has functional form $$u(x,t) = ...
3
votes
2answers
95 views

Solution of a specific PDE

I'm looking for a solution of the following PDE problem: $$\begin{cases} -\frac{\partial^2u}{\partial x^2}-2\frac{\partial^2u}{\partial y^2}=f&\text{on}~U,\\ u=0&\text{on}~\partial U ...
1
vote
0answers
42 views

Looking for numerical soution for a Non-local advection and diffusion equation with mass conserved

Here is the system: $\rho_t+(\rho v)_x=0$ and $v=f*\rho$(convolution). $\rho(x,t)$ is the density function and $f(x)$ is a function of $x$. The domain for $x$ is $[0,\infty]$. What I'm doing is to ...
2
votes
1answer
66 views

An imbedding question

Is it possible to say that $$ H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\hookrightarrow H_{0}^{1}(\Omega). $$ Precisely, I am dealing with this question: Is it possible to have the following estimate if we ...
2
votes
1answer
122 views

Want to show an operator is compact

With $V=L^2(0,T;H^1(\Omega))$, let $A:V \to V^*$ with $$\langle Au,v \rangle = \int_0^T \int_{\Omega} \nabla u(t) \cdot \nabla v(t).$$ I want to show that $A$ is a compact operator. So, one way to ...
7
votes
0answers
165 views

$W^{1,p} $ and $W^{2,p}$ Estimates.

In the beginning of section 4 in here the author says that one can easily adapt the methods in the preceding section to obtain $W^{1,p}$ estimate. I'm trying to do this. I think the following: the ...
3
votes
2answers
323 views

Eigenfunctions of Laplacian and orthonormal basis (with different inner products)

Suppose I have $L^2(\Omega)$ which has two inner products that are both norm-equivalent. The eigenfunctions of the Laplacian $\Delta$ we know forms an orthonormal basis of $L^2(\Omega)$ -- with ...