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

learn more… | top users | synonyms (2)

0
votes
0answers
21 views

Why this operator is not fredholm?

Define $f:S^{2}\to \mathbb{R}$ by $f(x,y,z)=z$. Let $D:=D_{\nabla f}$. As I learned from the following post this operator is not counted as a fredholm operator.( I did not underestand, ...
1
vote
0answers
23 views

Conceptual difficulty about eigenfunction expansions

I'm having a fairly large conceptual block on my understanding of eigenfunctions - I've tried out two different methods that yield embarrassingly different answers, and I would appreciate if someone ...
1
vote
1answer
26 views

Computing a certain partial derivative

I'm working through this PDE textbook and am trying to understand where I'm going wrong in reproducing a computation of a second partial derivative $u_{x_i x_i}$ in the text (for reference, ...
1
vote
1answer
35 views

Find a harmonic function on two concentric balls?

My attempt: I thought about using Poisson Integral formula since the area is two concentric balls. Then I get something like the following: $u(x)=\frac{1}{nw_nR}\int_{\partial ...
1
vote
1answer
51 views

Inhomogeneous Wave Equation Derivation

This is an assignment question which I've been working on to solve the inhomogeneous wave equation $u_{tt} - c^{2}u_{xx} = f(x,t)$. I separated the equation out into a system of two equations: $u_{t} ...
1
vote
1answer
39 views

The solution of $\Delta u=u^3$ with zero boundary values is identically zero

My question: My attempt: I tried to use the Representation using Green's formula: Since $u=0$ on the boundary and $f(x)=x^3$, then the formula becomes: $$u(x)=\int_\Omega y^3G(x,y)dy \quad ...
1
vote
0answers
23 views

Comparison of highly nonlinear parabolic PDE

Let $[0,T]$ be the time domain, and $I:=(-\pi,\pi)$ be the space domain. Consider a parabolic (4th order, nonlinear) PDE $$u_t=-A(u_x,u_{xx},u_{xxx})-B(u_x,u_{xx},u_{xxx},u_{xxxx}),\quad ...
0
votes
0answers
40 views

_How to prove $\triangle h=f$??

there! The following problem has been really bothering me for hours. My problem: How to prove $h(x)=\int_{R^n}\Gamma (x-y)f(y)dy$ is bounded and $\triangle h=f$ in $R^n$. My attempt: I found the ...
0
votes
0answers
9 views

Existence of a sequence of regular values converging to a given point

Suppose I have a function $f \in C^{1}(\mathbb{R}^{2}; \mathbb{R})$, $f \geq 0$, with compact support. Am I correct to think that the level sets of $f$ corresponding to regular values (i.e. values ...
0
votes
0answers
15 views

Regularity of the border of a set

How is regularity defined on the border of a set? Suppose I have an open subset $U \subset \mathbb{R}^{n}$ with border $\partial U$, what does it mean for $\partial U$ to be Lipshitz? $C^{1}$? ...
0
votes
2answers
53 views

Sobolev space on union of two open sets

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and ...
0
votes
0answers
39 views

Estimates on solution of pde

Let's consider the following pde in $\mathbb{R}^n$ $$\partial_t u=(i+\varepsilon)\Delta u,\,\,\,u(0,x)=u_0(x)$$ How to get the following estimate for its solution? $$\Vert u\Vert_2\leq C_\varepsilon ...
0
votes
0answers
25 views

I need a general solution of the following PDE:

$$\frac{\partial}{\partial t} F(t, x) = xF(t,g(tx)), F(0,x)=1=F(t,0) $$ where $g(x)$ is given. In fact, I need the case $g(x)=x.$ This PDE comes from an integral equation $$ F(t,x)=1+x \int_0^t F(s, ...
1
vote
2answers
37 views

heat equation with fourier series

Original PDE $$T_t=\alpha T_{xx}$$ I need to solve this equation numerically and analytically and compared them. I've already done the numerical part. But I need to solve it analytically now. Given ...
0
votes
1answer
23 views

Harmonic functions that uniformly convergent?

Let $u_k$ be continuous on $\overline\Omega$, $u_k$ harmonic in $\Omega$. Suppose $u_k|\partial\Omega$ converge uniformly. Then $u_k$ converge uniformly in $\Omega$. The hint is using Maximum ...
0
votes
1answer
42 views

Poisson's Equation in a Disk

How can we solve Poisson's equation in a disk in plane polar coordinates?: $$ \nabla^2 \phi = u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2} u_{\theta \theta} = f(r, \theta)$$ (My attempt): We know that ...
0
votes
1answer
31 views

If $F''(x) = -\lambda F(x)$ satisfies $F(L)F'(L) \leq F(0)F'(0)$, show that $\lambda \geq 0$.

Consider the function $F \in C^{2}([0,L])$ which satisfies the eigenvalue problem $$F''(x) = -\lambda F(x)$$ and suppose that $F$ satisfies the following constraints on its boundary values ...
0
votes
0answers
15 views

Energy conservation on a fixed boundary - lower dimensions?

I'm having trouble understanding a proof given in my notes about energy conservation for the wave equation when the boundary values of the function $u$ do not change with time. So we have $$ u_{tt} ...
2
votes
0answers
27 views

Why study physical differntial equations in$\mathbb{R}^n$

Studying equations such as the heat equation and the wave equation in n$\le$3 dimensions makes sense to me as these are physical processes. I can also justify studying PDEs in $\mathbb{R}^n$ because ...
-1
votes
0answers
26 views

Converting partial DE to integral Equation [closed]

Can anybody help me solving the below problem: What would be the functional corresponding to the following problem: $$ \frac{\partial ^{2}u}{\partial x^{2}}+ \frac{\partial ^{2}u}{\partial y^{2}} = ...
1
vote
1answer
27 views

Integral of homogeneous partial differential equation

From the book "Radio Occultations Using Earth Satellites" by William G. Melbourne: From Calculus of Variations a necessary condition for stationarity is that the ray at all points must satisfy ...
1
vote
0answers
23 views

Motivation for weak solution of a PDE (initial condition)

When looking at a (nonlinear degenerate) PDE like one defines as weak solution as Now I wonder about (2.1). The deduction of (2.2) is ok to me, as I can use the standard way with partial ...
1
vote
2answers
22 views

Energy for Inhomogeneous Heat Equation

Suppose $V(x,t)$, $x\in \mathbb{R}^n$, $t\geq 0$ is continuous such that $V(x,t)\geq \epsilon >0$. Now I want show that for any solution $u$ of the $\partial_t u+\Delta u + V(x,t)=0$, the energy ...
2
votes
0answers
13 views

stationary stokes problem - inf-sup

I want to show that $\inf_v \sup_{p} \int_{\Omega} \vert \nabla v \vert^2 + p \nabla \cdot v \, dx$ (where $v \in W_2^1(\Omega)$ and $p \in L^2(\Omega)$) is equivalent to the minimization of the ...
0
votes
0answers
33 views

Norm independent solution for partial differential eqution

We are looking for raidal (norm independent) solutions for $u_{xx}=u_{yy}$. I have solved this equation and get that: u(x,y)=$f(x+y)+g(x-y)$. Now since the solution should be norm independent ...
0
votes
2answers
40 views

How is this boundary condition $\lim_{t \rightarrow -\infty} f(x,t) = y_0(x)$ for a PDE called?

If you have a diffusion equation $\partial_t f(x,t) = \partial_x^2 f(x,t) $, where $(x,t) \in [0,a] \times \mathbb{R}$ and then you say $\lim_{t \rightarrow -\infty} f(x,t) = y_0(x)$, how do you call ...
1
vote
0answers
19 views

Fourier Series Problem

I am having trouble with this problem: For the following problem, sketch $f(x)$, the Fourier Series of $f(x)$, the Fourier Series and Fourier Cosine Series of $f(x)$. $f(x) = 1$. I think the ...
2
votes
0answers
30 views

How to derive the cigar soliton solution to the Ricci flow equation?

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form $$ {\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}} $$ I am ...
2
votes
1answer
37 views

Proof of reflection principle for harmonic functions

** My attempt: Hi, there! I have known how to prove the above statement when $u\in C^2(U)$, however, I have question about proving the above statement. Because it is $u\in C^2(U^{+}) \cap ...
0
votes
1answer
11 views

intuitive question of pde: The odd reflection is weak solution of this equation in the weak sense?

Consider $B_1 = B(0,1)$ the unitary ball of $R^n.$ Denote $B^{+} = \{ x \in B_1 ; x_n > 0\}$ and $B_{ - } = \{ x \in B_1 ; x_n \leq 0\}$. Let $u \in L^{\infty}_{loc}(B^{+}) \cap W^{2,2}(B^{+}) \cap ...
1
vote
1answer
32 views

Laplace equation, Taylor expansion

I couldn't find it anywhere, so I decided to write my question here: I have problems solving this equation: $$u_{xx} + u_{yy} = 4,$$ subjected to the conditions $$u(x,x)=2x^2, \quad u_x(x,x)=2x$$ ...
1
vote
0answers
13 views

Least squares problem equivalent to solving Poisson problem for graph embedding given edge lengths

Suppose we are given a set of edge lengths $\{e_j\}$ and want to recover vertex positions $\{x_i\}$ of a valid graph embedding that realizes the given edge lengths as best as possible. More precisely, ...
1
vote
1answer
41 views

Partial differential equation $3u_y+u_{xy}=0$

I am only starting my PDE course and I have problems solving this easy equation. $$3 \frac{\partial u}{\partial y} + \frac{\partial ^2 u}{\partial x \partial y} = 0$$ Here's what I've tried: ...
1
vote
1answer
39 views

Exercise on discontinuous coefficients in first order pde

I'm trying to solve the following exercise on first order pde with discontinuous coefficients, which I've found online. It consists in giving an unambiguous meaning to the following equation ...
1
vote
0answers
23 views

On Yau's (and Schoen's) proof of the positive mass theorem

I would like to face the proof of the positive mass theorem by Yau and Schoen. I have a Bsc in Mathematics and a Msc in Theoretical Physics and I'm preparing a PhD interview-challenge where I have to ...
1
vote
1answer
20 views

Partial differential equation, known solution

Knowing that any solution of $y \frac{\partial ^2 u}{\partial y^2} + 2 \frac{\partial u}{\partial y} = \frac{2}{x}$ is of the form $u(x,y) = \frac{\varphi(x) + y^2}{xy} + \psi(x)$, where $\varphi, \ ...
1
vote
1answer
29 views

I need a counterexample can prove that sequence of harmonic functions may not convergences to a harmonic function

Firstly, we know that if the sequence is uniformly convergences to a function then it must be harmonic. As the title,just given the sequence convergences to function, can we get it will be a ...
3
votes
2answers
66 views

A harmonic function bounded from below is constant

I am learning PDE on myself as a beginner. It takes me like several hours to finally think out this proof. However, I feel something not right about my proof, especially choosing "$R$" part, it ...
3
votes
1answer
73 views

Maximum principle for harmonic functions on unbounded domain

I am learning PDE by myself now. I am considering converted the problem to bounded domain to use the strong maximum principle. My attempt: Using the $\lim u(x)=0$, then exists $\epsilon$ and $N$, ...
1
vote
1answer
24 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
2
votes
1answer
40 views

Why does an eigenvalue expansion 'work' for PDEs?

I understand the logic and rationale behind using a series of eigenfunctions to represent general solutions to simple partial differential equations with prescribed boundary values, such as the ...
0
votes
0answers
14 views

finding the solution space of helmholtz equation with mixed boundary?

Let $\Omega\subset \mathbb{R}^2$ be a bounded set and $\Omega' \subset \Omega$ such that $\partial\Omega \cap \partial\Omega' = \emptyset$ $$ \left\{ \begin{align} k^2 u + \Delta u &= 0 \quad ...
0
votes
0answers
19 views

A special property of harmonic function

In $\mathbb{R}^n, $ suppose $u$ is a harmonic function in $B_1 (0)$. If $ 0\leq a\leq 1$ , then how to show that $$\int_{S^{n-1}} u(a^2 \omega)u(\omega)d\sigma_{\omega}=\int_{S^{n-1}}u^2(a\omega) ...
0
votes
1answer
30 views

Mollification of $L^{\infty}$ functions

We know when $1\leq p<\infty$ , the mollification function $f^{\epsilon}=\phi_{\epsilon}*f$ for $L^{p}(R^n)$ functions converge to $f$ in $L^{p}$ norm, when $p=\infty$ it might be wrong. But who ...
1
vote
0answers
17 views

Drums and wave equation, a clarification of an assertion

I have read somewhere something that sounds like that: 'Given the shape of a drum , thanks to well-known formulas, we can deduce all its vibration frequencies and so the sound that it is capable to ...
0
votes
0answers
12 views

Solve a PDE in the distribution sense.

I want to solve (in the distribution sense) this equation: $ x^{2} u= \delta_{0}$. I tried to use the variational form to deduce u but I get stuck. Can someone help? thanks.
1
vote
1answer
22 views

Is it a Sobolev function?

Let $U=(-1,1)\times(-1,1)$. Define $$ u(x)=\begin{cases}1-x_1 \;\;\text{if}\;\;x_1>0,|x_2|<x_1 \\1+x_1\;\;\text{if}\;\;x_1<0, ...
2
votes
3answers
30 views

Confused solving quasilinear PDE

The equation to solve is: $\displaystyle x\frac{\partial u}{\partial x}+ \displaystyle y\frac{\partial u}{\partial y}+ \displaystyle z\frac{\partial u}{\partial z}= \displaystyle xyz$ Applying the ...
2
votes
1answer
43 views

Parametrisation of boundary conditions for a quasilinear wave equation

Exercise 12.6.10 from the book Applied Partial Differential equations (Haberman) seems to be distinctly different from the other exercises. It is formulated thusly: Solve $\frac{\partial ...
1
vote
0answers
18 views

Help with the definition of weak solution

I've just begun studying PDE. In order to prove that $w \in C^1(D)$ is a weak solution of \begin{equation} \begin{cases} \Delta\:g+ \lambda \:g=0\quad {\rm in}\;D \\ g=0\quad {\rm on} \; \partial ...