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

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6
votes
2answers
70 views

Decay of $H^1(\mathbb{R}^n)$ functions

Is it true (is there a commonly known theorem) that says: $f \in H^1(\mathbb R ^n)$ $\Rightarrow$ $\displaystyle \lim_{|x| \to \infty} f(x) = 0$ pointwise (where $H^1$ denotes the Sobolev space ...
2
votes
3answers
320 views

Thermodynamics for math majors

I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis. I'd like to do some reading into the field of thermodynamics. ...
0
votes
1answer
19 views

Solving a PDE for walkSat problem.

Let $p(d)= \dfrac{1}{k}p(d-1)+\dfrac{k-1}{k}p(d+1)$ Assuming $p(0)=1$ and $p(\infty)=0$, show that $p(d)=(k-1)^{-d}$. What method do you use to solve these equations? I'm reading it out of a ...
2
votes
1answer
48 views

Asymptotic behaviour of solutions to elliptic PDE

Let $u$ be a solution (in the distributional sense) of $$ \Delta u = \delta_r $$ on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$. Let $w$ be a solution of $$ Aw = \delta_r $$ where $A = ...
0
votes
0answers
39 views

Small Question about a heat conduction problem

Find the solution $y(x,t)$ in $0<x<3$ and $t≥0$ for the following heat conduction problem: $2y_t=y_{xx}$; $y_x(0,t)=y_x(3,t)=0$; $y(x,0)=g(x)$ Where $g(x)$ is a generic function. ...
1
vote
2answers
47 views

Inequality for solution of boundary-value problem

Let $\varphi(x,t)$ be a sufficiently smooth solution of the problem $$\left\{\begin{array}{rcll} \frac{\partial\varphi}{\partial t}(x,t) - \frac{\partial^2\varphi}{\partial x^2}(x,t) & = & ...
2
votes
1answer
74 views

Difference Quotient Proof

Theorem: Let $u \in W^{1,p}(U)$ and let $V \subset \subset U$ (I.e. there is a compact set containing $V$ that is in $U$). Then for $1\le p <\infty$ there exists a constant $C$ such that for $0 ...
1
vote
1answer
217 views

Parabolic PDEs: Boundary conditions

I'm working through Pinchover and Rubinstein's "Introduction to Partial Differential Equations" and am trying to understand the motivation for studying Sturm Liouville problems. To this end, I am ...
5
votes
0answers
146 views

Operator completly continuous

For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$ and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
5
votes
0answers
91 views

Solution to $\Delta_g u = \delta-1$ on a 2-sphere.

Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
1
vote
1answer
241 views

How to determine if the sums and products of harmonic functions is also harmonic?

Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + ...
1
vote
1answer
64 views

compare of eigenvalues $\lambda_1(D_a)$ and $\lambda_1(D_c)$.

Let $f(x)$ be a smooth function on $[-1,1]$, such that $f(x)>0$ for all $x\in(-1,1)$,$f(-1)=f(1)=0$. consider $\gamma\subset\Bbb{R}^2$ the graph of the $f(x)$. Let $T_a$ the symmetry with respect ...
2
votes
1answer
53 views

Help with PDE/Green's formula question.

Let $D$ be a bounded region. For the problem $\Delta u+au-bu^2 = 0$ in $D$$u=0$ on the boundary of $D$, show that there are no positive solutions if $a<\lambda_1$ (which is the smallest ...
2
votes
1answer
68 views

Trouble with starting PDE energy integral question.

Problem: We have $$u_{tt} = c^2\Delta u$$ on on $D\times (0, \infty)$ for some bounded region D in three dimensions, with $u=0$ on $\partial D \times (0, \infty)$. Show that the energy integral, $$ ...
3
votes
2answers
410 views

Prove an identity for the continuous integral solution of the conservation law

This is an exercise in Evans, Partial Differential Equations, page 164, problem 13 Assume $F(0) = 0, u$ is a continuous integral solution of the conservation law: $$ \left\{ \begin{array}{rl} u_t + ...
2
votes
1answer
103 views

Mean Value Property

I'm currently studying the theory of PDEs and, in particular, harmonic functions. I've been given this question: Show that if $u:(a,b) \rightarrow \mathbb{R}$ is continuous, and satisfies the ...
4
votes
1answer
172 views

“Penalty method” to approximate solutions of a variational inequality

The following is Problem 3 from Chapter 9 of Evans's book on PDE, 2nd edition. (Penalty method) Let $\varepsilon>0$ and define $$\beta_\varepsilon(z)=\begin{cases} 0 & z \ge 0 \\ ...
1
vote
1answer
163 views

the first eigenfunction of Dirichlet problem

Let $\Omega$ be a bounded planar domain which has a axis of symmetry and $T:\Bbb{R}^2\longrightarrow\Bbb{R}^2$ symmetry with respect to this axis. Let $u_{1}(x)$ be the first eigenfunction of ...
2
votes
1answer
150 views

Divergence free, smooth functions on unit circle.

I need to construct a divergence free, smooth, vector function on a unit circle such that $ \mathbf{u} = (u_1,u_2) = (0,0) $ on $\partial B$, and $\int\limits_B u_i \neq 0, \ i=1,2$. I was able to ...
1
vote
0answers
63 views

A damped wave equation with non null BC

Do exist any physical interpretation for the following wave equation with non null boundary conditions $ y=g \ne 0 $ on $\partial \Omega :$ $$y_{tt}=\Delta y + a(x) y + b(t)y_t, \;\mbox{in}\; ...
3
votes
1answer
2k views

How to solve partial integro-differential equation?

Suppose the following partial integro-differential equation for a function $u(x,t)$ with $t\geq0$, $x \in [0,L]$: $\partial_t u = \partial_{xx} u + f(u,\lambda)$ $\lambda = B\left(u_0 - \int_{x=0}^L ...
2
votes
0answers
115 views

What are the properties of this kind of Fourier transform when using it to solve Wave equqtion with Robin boudary data?

I am facinated by the following problem (Which is found in Budak & Samarskii's Book Collection of Problems on Mathematical Physics, about on Page 53.): Solve the following problem by using ...
0
votes
1answer
74 views

Doubt on how to proceed with this PDE

Solve $$u_t = ku_{xx} + xe^t$$ where $0<x<\pi$ and $t>0$ $$u(0, t) = u(π, t) = 0 \quad , \quad t>0$$ $$u(x,0)=0 \quad, \quad 0<x<\pi$$ Attempt: I began by separating variables ...
3
votes
3answers
161 views

If $\frac{\partial \varphi}{\partial x}=f(x,y),\frac{\partial\varphi}{\partial y}=g(x,y)$, what is $\varphi$?

Suppose we have a real-valued function $\varphi(x,y)$ such that $$ \frac{\partial \varphi}{\partial x}=f(x,y)\quad\text{and}\quad\frac{\partial\varphi}{\partial y}=g(x,y) $$ for some functions $f$ and ...
0
votes
1answer
63 views

transformation from pde to a simpler heat equation

Please give me hints to transform the following PDE to a simpler heat equation: $$\dfrac{\partial u(t,x)}{\partial t}+k(\rho-\ln x)x\dfrac{\partial u(t,x)}{\partial ...
0
votes
1answer
203 views

Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
2
votes
1answer
127 views

Visualize soliton solutions of a PDE

In trying to visualize soliton solutions of a PDE I faced this sentence: We now think of solitons as self-similar solutions, i.e., solutions which evolve along symmetries of the flow. Question 1: ...
2
votes
0answers
117 views

Convection-diffusion PDEs and time-dependence

I have the following equation: $$ \frac{\partial u}{\partial t} + \frac{\partial}{\partial x}(Cu) - \frac{\partial}{\partial x}\left(D\frac{\partial u}{\partial x}\right) = f(x,t) $$ And I've ...
19
votes
1answer
561 views

Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
1
vote
2answers
71 views

Is $1/z$ differentiable on $\mathbb{C}\setminus\{0\}$?

The way I usually solve these kind of questions is by using the Cauchy Riemann Equations... $h(z)=\dfrac{1}{z}$ $=\dfrac{1}{x+iy}$ $=\dfrac{1}{x+iy} \dfrac{x-iy}{x-iy}$ ...
5
votes
2answers
116 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 ...
2
votes
0answers
88 views

Can we say approximation of Sobolev function by a smooth function is inspired from “Weierstrass approximation”? [closed]

As we see that the idea of approximation of a function in Sobolev space by a smooth function looks like an abstraction of Weierstrass approximation theorem..
2
votes
1answer
157 views

question about wave equation

Dalambert's formula for the following BVP $$\hspace{35mm}u_{tt}=C^2u_{xx} ~~-\infty <x<\infty, t>0$$ $$u(x,0)=\phi(x)$$ $$u_t(x,o)=\gamma(x)$$ is that ...
1
vote
3answers
129 views

Non homogeneous Heat Equation PDE

$$A_t-A_{xx} = \sin(\pi x)$$ $$A(0,t)=A(1,t)=0$$ $$A(x,t=0)=0$$ Find $A$. I know I need to find the homogeneous and particular solutions. Im just not sure on this PDE.
3
votes
1answer
563 views

How to derive the solution of an three-dimensional wave equation with Cauchy data

Recently I am thinking the solution of the following 3-dimensional wave equation with Cauchy data: \begin{align*}&\frac{\partial^2 u}{\partial t^2}=4 \Big(\frac{\partial^2 u}{\partial ...
2
votes
2answers
88 views

What happens when you change space of test functions associated with weak derivatives?

Recall that $u \in L^2(0,T;H^1)$ has weak derivative $u' \in L^2(0,T;H^{-1})$ iff $$\int_0^T uv' = -\int_0^T u'v$$ holds for all $v \in C_0^\infty(0,T).$ What happens if we only require that this ...
0
votes
1answer
58 views

Partial Differential limits question.

I have a PDE that I solve to be: $$u(x,t) = f(x+at) + g(x-at)$$ I need to apply initial conditions, $u(x,0)=r(x)$ and $u_t(x,0)=s(x)$ From this I get: $$f(x)+g(x)=r(x)$$ and $$a(f'(x)-g'(x) = ...
0
votes
2answers
139 views

$\Delta u=0$ inside the quarter circle

Find the general solution (as a series) for $\Delta u=0$ inside the quarter circle $\left\{(r, \phi): r<2,\, 0<\phi<\dfrac{\pi}{2} \right\}$ with $u(r,0)=\dfrac{\partial u}{\partial ...
2
votes
1answer
397 views

canonical form for hyperbolic PDE $y^2u_{xx}+2xyu_{xy}+u_{yy}=0$?

How can the PDE $y^2u_{xx}+2xyu_{xy}+u_{yy}=0$ be reduced to canonical form in its hyperbolic region, namely $|x|>1,y\neq0$? I know the required substitution $(\xi(x,y),\eta(x,y))$ should be given ...
2
votes
1answer
45 views

The extension of smooth function under the restriction of its Laplacian

$u$ is a smooth bounded function on $\Omega-\{0\}$ where $\Omega$ is an open neighborhood of $0$ in $\mathbb R^n$. If $\Delta u$ is a bounded function on $\Omega-\{0\}$, then can we extend $u$ to be a ...
1
vote
1answer
88 views

PDEs: subsequence converges to solution, so whole sequence does too

Suppose we want existence of a function $u$ for the PDE $$(\frac{d}{dt}u,v) = b(u,v)$$ for all $v$ in a test space. Sometimes in PDE you have use a Galerkin approximation, so say $u_n$ is a sequence ...
1
vote
1answer
95 views

solution to heat equation in a particular case

$$\frac{\partial u}{\partial t}(t,x)-\frac{1}{2}\frac{\partial^2 u}{\partial x^2}(t,x)=0,\ \ t>0, x\in\mathbb{R}$$ $$u(0,x)=\max(x,0)$$ $$\frac{\partial v}{\partial ...
4
votes
1answer
195 views

Black-76 pde hedging argument wrong

I want to obtain the PDE for the Black-76 model. I believe it has to be the following PDE: $$\left(\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}F^{2}\frac{\partial^{2} V}{\partial ...
3
votes
1answer
103 views

Regularity for solutions of $-\operatorname{div}(g(|\nabla u|^2)\nabla u)=f$.

Let $\Omega\subset\mathbb{R}^N$ and suppose that $g\in C^{1,\alpha}(\mathbb{R},\mathbb{R})$, $f\in C^{0,\alpha}(U)$ for every open $U$ with $\overline{U}\subset\Omega$, $\alpha\in (0,1)$ and $g\geq ...
1
vote
1answer
132 views

laplace equation in a rectangle with boundary condition

$u_{xx}+u_{yy}=0 \quad in \quad the \quad rectangle \quad 0<x<a \quad 0<y<1$ $u=0 \quad on \quad y=1$ $u=j(y) \quad on \quad x=0 $ $u_y +u=0 \quad on \quad y=0$ $u_x=0 \quad on ...
3
votes
1answer
100 views

Weak solution of a non-linear problem with Lipschitz functions

I'm trying to solve the problem 9.5 in Evans PDE book. The statement goes as follows: Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a $\Lambda$-lipschitz bounded function with $f(0)=0$ and ...
2
votes
1answer
211 views

The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$

I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
3
votes
2answers
110 views

Elliptic regularity - nonlinear case

Let's say I have a weak solution $u \in H^1(\Omega)$, $\Omega \subset \mathbb{R}^n$ open, to the equation $$ \Delta u = e^u, $$ let's also assume that $e^u \in L^\infty(\Omega)$. Does it follow that ...
6
votes
2answers
233 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 ...
3
votes
0answers
52 views

Regularization of solutions from a quasilinear equation.

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Let $-\Delta_p :W_0^{1,p}\to W^{-1,p'}$ be the $p$-Laplace operator, i.e. $$\langle-\Delta_pu,v\rangle=\int_\Omega ...