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

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118 views

On pointwise bounded subsequences of a convergent sequence in $L^p$

When trying to rigorously formulate a proof presented to me during a PDE seminar I came across the following difficulty: Let $(u_n) \subset H_0^1 (\Omega)$ be a bounded sequence such that $u_n \to u$ ...
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174 views

Solving $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=c$

Question: let $a,b,c$ be positive constants. Find $u=u(x,y)$ if is satisfies the partial differential equation $$\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=c$$ and the ...
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105 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
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50 views

Please explain this notation of mapping into a set and product space (related to Sobolev spaces)

So does this mean that I can say that, for example, $\gamma \frac{\partial u}{\partial \nu}$ has a unique continuous extension as an operator from $W^s_p(\Omega)$ onto $W^{s-1-{\frac 1 ...
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79 views

Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$

I have a problem: For $\Omega$ be a domain in $\Bbb R^n$. Show that the function $u(x)=1 \in W^{m,\ 2}(\Omega)$, but not in $W_0^{m,\ 2}(\Omega)$, for all $m \ge 1$. ...
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219 views

(Obvious?) Half-Space Poisson Kernel Estimate

In Stein's Singular Integrals and Differentiability Properties of Functions, Theorem 1 (a) of Chapter 8 (pg. 197) he makes the claim without proof that the Poisson Kernel for the half-space ...
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42 views

First order linear pde with additional partial derivative constraints

Suppose we wish to solve the first order pde for the unknown function $f(x,y)$ $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=c(x,y)\Big(a(x,y)+b(x,y)\Big)$ We assume that the ...
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553 views

Derive the equation of motion from Lagrangian of a particle moving in an electromagnetic field

I really don't even know where to start with this question any help would go very very far. Thank you. A particle with charge $q$ moving in an electromagnetic field is described by the Lagrangian ...
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313 views

Neumann Problem Solutions

I want to prove that two solutions to a Neumann problem differ at most by a constant. $$\bigtriangledown^2=f \ in \ D$$ $$\frac{\partial{u}}{\partial{n}}=g \ on\ B$$ I don't know how to approach the ...
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143 views

Linearization by freezing the coefficients of the main part of the PDE

Let $\Omega\subset C^0$ a bounded domian in $\mathbb{R}^2$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a non negative classical solution of $$ ...
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56 views

Solving a PDE - do you have an idea?

Do you have an idea how to solve $$ v_{\xi\eta}=\frac{1}{2} v_{\xi}\cdot\xi? $$ First I thought of using $$ v_{\xi\eta}=v_{\eta\xi}, $$ substituting $z:=v_{\xi}$ and then getting $$ ...
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68 views

differentiability/holomorphicity of family of bounded operators

Edit: It seems I made a mistake in the statements on differentiability. I will replace weak differentiable implies strong differentiable with weak continuously differentiable implies strongly ...
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38 views

Looking for a basis of $L^2$ with this special property

The setup. Let $\mathbb{T^2}$ denote the two-dimensional torus, i.e. $$ \mathbb{T}^2 \simeq [-\pi,\pi)^2 $$ induced by identifying opposing faces of $[-\pi,\pi)^2$. Note that $$ L^2(\mathbb{T^2}) ...
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76 views

To show that $ J(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1} $ is not bounded above for $1 < p < 2^*-1$

For a bounded $ \Omega\subset\mathbb{R}^n $ with smooth boundary, and for $ 1 < p < \frac{n+2}{n-2} = 2^* -1 $ where $ \frac{1}{2^*} = \frac{1}{2}-\frac{1}{n}$, I have the functional $ J : ...
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99 views

$u\in H^1(\Omega)$ implies $\Delta u\in H^{-1}(\Omega)$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary. Is it true that if $u \in H^1(\Omega)$, then $\Delta u \in H^{-1}(\Omega)$? Thanks
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183 views

How to solve $f\frac{\partial^2f}{\partial x\partial y} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y}$

I want to solve the following non-linear PDE: $$f\frac{\partial^2f}{\partial x\partial y} = \frac{\partial f}{\partial x}\frac{\partial f}{\partial y}.$$ I don't know much about solving PDE's, ...
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120 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 ...
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61 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 ...
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121 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) = ...
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52 views

Need a source for the following result: From $v \in H^1$ and $f$ Lipschitz it follows that $f(v) \in H^1$.

I am looking for a source of the following (or a similar) result: If $v \in H^1(\Omega,\mathbb{C})$ on a bounded domain $\Omega$ and $f: R(v) \to \mathbb{C}$ is Lipschitz continuous, then $f(v) ...
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283 views

Definition of Hölder space

I am wondering what is the definition for Hölder space $C^\gamma$ when $\gamma\in \mathbb{N}$. Let's take the underlying field $\mathbb{R}^d$. Is it $$ C^\gamma = \{f:\ f\in C^{\gamma-1}\}\cap \{f: ...
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188 views

An example of Kirchhoff equation

I'm studying about a simply kind of Kirchhoff equation in one-dimensional that means $$\begin{cases} ...
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622 views

Relationship between Turing bifurcation, saddle-node bifurcation, and Hopf bifurcation?

Quoting from http://jxshix.people.wm.edu/2009-harbin-course/mississippi-bifurcation-2.pdf a Turing bifurcation occurs when for an ODE and related PDE $u' = f(u,v), v' = g(u,v)$ $u_t = d_1 \nabla ...
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63 views

spherically symmetric configurations

$$\Delta S -S +S^3=0$$ How this Differential equation can be written in this form: \begin{equation} \frac{d^2S}{d\rho^2}+\frac{D-1}{\rho}\,\frac{dS}{d\rho} -S+S^3=0 \end{equation} Which is ...
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461 views

The harmonic conjugate of $\Im e^{z^2}$?

It is obvious that $e^{z^2}$ is analytic, right? So the harmonic conjugate of $\Im e^{z^2}$ is $\Re e^{z^2}$, isnt' it? However, the solutions manual I'm consulting gives the answer as $\Im ...
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72 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... $h(z)=\dfrac{1}{z}$ $=\dfrac{1}{x+iy}$ $=\dfrac{1}{x+iy} \dfrac{x-iy}{x-iy}$ ...
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103 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 ...
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121 views

Laplace Equation PDE

$$U_{XX}+U_{YY}=1 $$ in the annulus $ a<r<b$ with u vanishing on both parts of the boundary r=a and r=b What I have done is that $u_{xx}+u_{yy}=0$ has only 0 due to the fact that u is vanishing ...
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85 views

Can someone explain to me Feyman Kac and walk through an example?

I kind of understand what needs to be done to convert an SDE to a PDE but I don't understand why we're allowed to do it. What is the generator? ie: given $dS(t) = rS(t)dt + \sigma S(t)dB(t)$ we get ...
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163 views

Existence of smooth function with given compact support

Let $K$ be a compact set in $\mathbb{R}^n$. Does there exist a smooth function $\phi$ such that $0<\phi\leq 1$ on $K$ and 0 outside of $K$
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377 views

Cannot understand method of characteristics

$u_t+uu_x =0$ $u(x,0)\equiv u_0(x)=\begin{cases} 0 & x<0 \\ 1 & x>0 \end{cases}$ I want to parametrise $u(x(s),t(s))$. This is the first thing that is conceptually quite difficult to ...
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129 views

Laplacian of the potential function

Given a potential function $$\Gamma(x)=\frac{1}{(n-2)|S^{n-1}||x|^{n-2}}$$ where $n\ge 3$ is an integer, $|S^{n-1}|$ is the volume of the sphere, $x$ is in $n$ dimension. Is it true that $$-\Delta ...
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210 views

partial derivative of $L^2$ norm?

In the chapter on energy methods for partial differential equations I saw the following: $$\frac{d\|u\|_2^2}{dt}=(u,u_t)+(u_t,u)=\cdots$$ So, why we can't just write ...
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253 views

Partial Differential Equation about Rotation question

Consider the constant coefficient PDE: $$a_{11}u_{xx}+2a_{12}u_{xy}+a_{22}u_{yy}+b_{1}u_{x}+b_{2}u_{y}+cu=0$$ Show that the only ones that are unchanged under all axis-rotations (rotation invariant) ...
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316 views

Solving Heat Equation PDE

I need some help on solving this heat PDE : Question : Consider a bar length L. The face at x=0 is insulated so that the heat flow across is zero, and the face at x=L is held at temperature u=0. The ...
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73 views

Writing $\int_\Omega \Delta u \Delta v$ in a nicer way?

Is there a way to write $\int_\Omega (\Delta u)^2$ or more generally, $\int_\Omega \Delta u \Delta v$ more nicely (possibly after integrating by parts)? I want something like $\int \nabla f\cdot ...
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146 views

Partial Differential Equation

Please help me solving the following PDE: $\partial_t f=\sin (t)\,\partial_x f+\lambda \,\partial_{xx} f$ with initial condition $f(x,0)=1$ for all $x$
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281 views

Solving $u_t=4u_{xx}$

How do we solve the folllowing diffusion problem? $u_t=4u_{xx}$ $u(0,t)=0$ $u(3,t) = 0$ $u(x,0)=\sin(2\pi x/3)-2\sin(\pi x)+7\sin(5\pi x/3)$
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78 views

Question about derivatives of complex-valued functions

For $ z \in \mathbb{C}, t \in \mathbb{R}, \\f : \mathbb{C} \times \mathbb{R} \to \mathbb{C}, \\a : \mathbb{C} \times \mathbb{R} \to \mathbb{R}, \\b : \mathbb{C} \times \mathbb{R} \to \mathbb{R}$ And ...
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163 views

Solving a PDE possibly with method of characteristics or other methods

For a PDE $$(x-y^{2}) u_{x} + u_{y} = 0$$ I've tried to use method of characteristics. But I've failed to do so. It was because of the term $x-y^{2}$; I don't know how to integrate this on the ...
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201 views

Mean Value property for harmonic functions on regions other than balls/spheres

Let $u$ be a harmonic function on $\mathbb R^n$. We know that if $B$ is a ball centered at $x$, then $$u(x) = \frac{1}{|B|} \int_B u(y) dy = \frac{1}{|\partial B|} \int_{\partial B} u(z) dS(z).$$ I ...
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751 views

Solve the 1-D Heat Equation with the given boundary values and initial conditions

Solve: $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+4\frac{\partial u}{\partial x}+2u$ for $0<x<\pi$, $t>0$ with boundary conditions $u(0,t)=u(\pi,t)=0$ for $t>0$ and ...
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471 views

Wave Equation - like 4th Order PDE

How does one solve a fourth-order PDE of the form $\frac{\partial^4y}{\partial x^4}=c^2\frac{\partial^2y}{\partial t^2}$? It looks like a one dimensional wave equation, but I'm unfortunately very bad ...
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287 views

How to show that a given function is a solution for a partial differential equation.

Let $g : \mathbb R \to \mathbb R$ be a differentiable function and let $f(x,y)=x^ng(\frac {y}{x})$, where $n\in\Bbb Z^+$, show that $f$ is a solution to the PDE $$x\frac{\partial f}{\partial ...
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141 views

One-dimensional solution to Advection-diffusion equation

The advection-diffusion equation was $\frac{\partial c}{\partial t} = \nabla \cdot (D\nabla c) - \nabla \cdot (vc) + R$ where $c$ is a scalar (the concentration) and $v=(v_1 ,v_2 ,v_3 )$ is a vector ...
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71 views

Lebesgue integration for $u \in C^{\infty}_c$

Let $u \in C^{\infty}_c(\Bbb{R}^d)$, where $C^{\infty}_c(\Bbb{R}^d)$ is the family of infintly differentiable functions with a compact support. Is $u$ in $L^2(\Bbb{R}^d)$? I think that $u$ is in ...
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380 views

Numerical Solution of Systems of PDE

Could someone give me some reference to the Numerical solution of a System of PDEs of following type.. (which also encompasses strongly elliptic system of PDEs) in 2D or 3D. $$\left\{ ...
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202 views

Implicit function theorem and PDE; do we get uniqueness?

Please see this page: The implicit function theorem: A PDE example. In the implicit function theorem they quote, uniqueness is not mentioned. But the inverse function theorem (which is equivalent to ...
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691 views

Does solve PDE by combination of variables always cannot find the general solutions?

Combination of variables is the technique that reducing the PDE to one independent variables (i.e. become ODE) by introducing a suitable change of variables. But the general solution of a PDE should ...
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307 views

Determine characteristics for the method of characteristics

I'm trying to understand the method of characteristics to solve first-order PDEs. As an example in his course, my professor solve this PDE for $u(x,y)$: $$x\frac{\partial u}{\partial ...