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

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A question on the well-posedness of of p-Laplacian

Can it be shown that the problem \begin{eqnarray} -\Delta_{p} u &=& f(u),\nonumber\\ u|_{\partial\Omega} &=& g, \end{eqnarray} well-posed similar to the case when $p=2$?.
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2answers
30 views

End-of-semester presentation idea on PDE

I am supposed to give a 30-minute class presentation on any PDE subjects as end-of-semester project. Do you have any pet subject you would love to suggest? I have very little applied maths in my ...
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1answer
27 views

For which $p$ the sequence $x^n$ converges in the Sobolev space $W^{1,p}(I)$?

I would like to know for which $p$ the sequence $u(n)=x^n$ converges in the Sobolev space $W^{1,p}(I)$. Is it true that converges only for $p=1$? I find out this looking for which $p$ the Sobolev ...
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28 views

Question about representation of the eigenvalues of second order elliptic operator

let $Lu:=-\text{div}(A\cdot\nabla u)$, where $A$ is symmetric. Eigenvalues of $L$ is $\lambda_1<\lambda_2<\cdots$. By definition. (If exists a nontrivial solution $w$ such that $Lw=\lambda w$, ...
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16 views

Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
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24 views

Existence of Green's Function for Dirichlet Problem

Does anyone know where I can find a proof of the existence of Green's function, $G(x,x_0)$ on any nice enough domain $\Omega \subset \mathbb{R}^n$? Edit: This is for Laplace's problem
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2answers
45 views

Fourier inverse of a function to get dirac

I'm trying to get the dirac function from a fourier inverse tranform: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iw(x-x_0)}dw$$ It is this last step I am stuck on to get the conclusion. Original ...
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15 views

Uniqueness of solution to linear PDE, 2nd order in time with convection term

I am having troubles proving the uniqueness of the zero solution to the following Cauchy initial value problem for $\rho:\mathbb{R}_{\geq 0} \times \Omega \rightarrow \mathbb{R}$ with $\Omega ...
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1answer
18 views

Integration of Fundamental Solution of Laplace's equation.

I am currently reading Evan's PDE and am getting hung up on many of the more "technical details". This question may be very basic (multivariable calculus). I am given that the fundamental solution of ...
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1answer
44 views

Poincaré type inequality

Consider a sequence of functions $f_n \in H^1(M)$, the first Sobolev space of a complete (possibly noncompact) Riemannian manifold. If we have the normalization $\Vert \nabla f_n\Vert_{L^2} = 1$, ...
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1answer
32 views

Confusion over $\not\equiv$ and $\neq$ when applying boundary conditions

Here $k, A_1, A_2$ are constants Although $A_1$cos$kx+A_2$sin$kx \not\equiv 0$, $A_1$cos$kx+A_2$sin$kx$ can equal zero at certain values of $x$ For example if $A_1=1, A_2=1, k=1$ then ...
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1answer
49 views

A trace inequality with epsilon in Sobolev spaces

We know the standard trace inequality: for a bounded domain with certain boundary regularity, there is a $C>0$ such that $$ \|Tu\|_{L^2(\partial \Omega)}\leq C\|u\|_{H^1(\Omega)}, \quad \quad u\in ...
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16 views

nonlinear version of strong maximum principle

page 31 Vazquez has a nonlinear version of strong maximum principle which designated for Quasilinear parabolic PDE. Roughly it says, if $u_0\le v_0$, then either we have $u\equiv v$ or $u<v$ for ...
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29 views

Solution to klein-gordon type equation

While studying physics, I ended up having to find solutions for the following partial differential equation: \begin{equation} \left[ \frac{1}{2}\left( \frac{\partial ^{2}}{\partial \alpha ...
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26 views

find the fourier transform of $xf(x)$ appended

I've seen the method in which you prove this fourier transform, but what if you don't recognize that $$xf(x) e^{i k x} = \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] $$ would I ...
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1answer
49 views

Solving Eikonal Equation

The problem is the following: I have the bidimensional eikonal equation with non-constant propagation: $u_x^2+u_y^2=u^2$ The goal is: i) To find the characteristic strips for the parametrization ...
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1answer
22 views

Applying Fourier transform to a gaussian

Let $$G_\beta(w) = e^{\beta w^2}$$ Now I get the process of applying a fourier transform (or inverse) to get a new gaussian: $$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$ but in doing the ...
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1answer
11 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
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1answer
37 views

Separation of variables for fourth order PDE

How do I solve: $$u_t = -u_{xxxx} + \pi^2u_{xx},$$ with BCs: $u_x(0,t)=u_x(1,t)=u_{xxx}(0,t)=u_{xxx}(1,t)=0$ and initial condition $u(x,0)=\cos(\pi x)$. We have been told that we can use ...
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1answer
22 views

Laplace operator

The question is that Derive a formula for $\Delta(\frac{f}{g})$ in terms of $f, g, \nabla f, \nabla g, \Delta f, \Delta g$. Naturally, I apply the rules of gradient and divergence, and yield ...
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1answer
10 views

Derivative of Bessel J series… Do I reindex my summation?

Okay, short question: what happens to my index upon differentiation and why? This is a small step in a larger proof I'm working on... Given the series representation of Bessel J $$J_n = ...
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2answers
24 views

finding the maximum of a second derivative of a function given its differential equation

Say I have $u(x)$ satisfying $\frac{d}{dx}(a(x)u'(x)) = 0$ with $u(0) = 0$ and $u(l) = 1$. Is there any way to find the maximum of $u''$, for example, without having to solve for the function $u$ ...
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1answer
163 views

An interesting semi-linear PDE problem

Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi linear equation $$ -\Delta u+c(u)=f\;\;\text{in}\;\mathbb{R^n} $$ where $f\in ...
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1answer
36 views

The bubble function

In the finite element method and more precisely the MINI element method in two dimensions, they use a function called the "bubble function" which is related to a triangle K of the space meshing and is ...
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20 views

Dam Break Problem

Is $u=0$, $h=1$ at every point where $x<-t$? $\frac{dx}{dt}=-1$ on $C_-$ and $\frac{dx}{dt}=1$ on $C_+$ only initially (at $t=0$) so how do we know that characterisitics intersect at every ...
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25 views

Why is this equation not a sturm-liouville equation??

The equation is $$\frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2} - V_o\frac{\partial u}{\partial x}$$ By seperation of variables we obtain for the x problem : $k\alpha'' ...
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1answer
15 views

Find the Fourier Transform of piecewise finction

$$f(x) = \begin{cases} 0 & |x|> a \\ 1 & |x|< a \end{cases}$$ I have most of the solution, I'm just faltering on obtaining the sin(ax) part of the solution, I'm missing an exponential ...
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17 views

For $f\in L^1_{loc} (\Omega)$, $f=0$ almost everywhere in $\Omega$ provided $\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$

I need to show that $f=0$ almost everywhere in $\Omega$ provided $$\int_{\Omega}f(x)\Phi (x)dx=0 , \forall \Phi \in C_{c}^{\infty}(\Omega)$$ Here is how I have decided to proceed. Suppose there ...
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19 views

Implicit solution to Method of Characterstics

If I have that, $\sqrt cu+v=$ constant along the characteristic lines $x+\sqrt c t=$ constant Where $u=u(x,t), v=v(x,t)$ and $c$ is constant. Why is it that $\sqrt cu+v=f(x+\sqrt ct)$ where $f$ is ...
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1answer
8 views

the diagonal angle between a theta and phi vector and the x axis as well as the derivative

Find w, the angle ∠rox. That is, the angle from the red vector r to the positive ...
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59 views

Show this scalar conservation law has a unique smooth solution.

Let $\epsilon>0$, consider the I.V.P of the viscous conservaion law $$\begin{cases}u_t+\partial_xf(u)=\epsilon u_{xx}\\ u(x,0)=u_0(x)\end{cases}$$ Show that there exists a unique solution $u\in ...
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38 views

laplace equation with boundary problems

I have the following problem: $$U_{{_x}_{x}} + U_{{_x}_{x}}=1$$ with neumann boundary on the left and right and dirichlet boundary on the top and botton. All boundary conditions are equal to zero I ...
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20 views

Boundary of Green's functions for solving PDE's

This has more to do with considering the boundary conditions of a Green's function when you are constructing it for a PDE. Are the boundary conditions of the GREEN's function always homogeneous or is ...
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22 views

Kolmogorov equations

I'm having a difficulty to understand the difference between the Kolmogorov Forward and Backward Equation in how they describe probability density rather then their mathematical formulation (I know ...
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1answer
41 views

A question on the continuity of a functional

Suppose $u \in L^{p}(\Omega)$, $\Omega$ is a bounded subset of $\mathbb{R}^n$. Let $q+1<p$ and $p \geq 2$. Is the functional defined by $v\mapsto\int_{\Omega}u^qv$ continuous over ...
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55 views

solving PDE with state-dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
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17 views

How to make a 9-point two-dimensional stencil for a elliptic operator?

I want use a finite difference schem to discretizate the elliptic operator: $$ \nabla \cdot \left( k(x, y) \nabla p\right), $$ where $k(x, y)$ a positive scalar function and $p$ is the unknow. We can ...
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24 views

proof of Wiener’s criterion

I'm in my first course of PDE and I need to investigate the proof of Wiener's Criterion for Laplace Equation which says, if $\Omega \subset \mathbb{R}^n$$(n>2)$ is a bounded domain and $\partial ...
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1answer
19 views

How do I determine the weight function for a regular Strum-Liouville problem given a set of boundary conditions?

I am given an eigenvalue problem of the form: $$\frac{d}{dx}[p(x)\frac{d\phi}{dx}]+q(x)\phi+\lambda\sigma(x)\phi=0$$ with boundary conditions:$$\phi(1)=0$$ $$\phi(2)-2\phi'(2)=0$$ In this case ...
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19 views

Show a certain modeshape satisfies Helmholtz equation with boundary conditions

I need to show that a mode shape $\psi(x,y,z)$ given by: $p(x,y,z) = \psi(x,y,z)e^{-jk_0\lambda z}$ where $k_0 = \omega_0/c_0$, $c_0$ is the sound speed, and $\lambda$ is an unknown, non-dimensional ...
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21 views

Solve for the Green's function and show is not symmetric even though dirac is.

$$\frac{dG}{dx} + G = \delta(x-x_0)\text{ with }G(x;x_0) = 0 $$ I am somewhat lost in what to do to turn this into a Green's function. My Prof went about it by simply using an integrating factor, but ...
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Generalization of Laplacian equation

Let $(r, \theta)$ be polar coordinate on $ \Bbb R^2 - \{0\} $. Expressing $\Delta u= u_{xx} + u_{yy}$ in polar coordinate we get $$ u_{xx} + u_{yy}= u_{rr} + 1/r u_r + 1/{r^2} u_{\theta \theta}$$ ...
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1answer
60 views

Solution of $u_t=u_{xx}+xu_x$

I've been asked to solve the problem \begin{equation} \left\{\begin{array}{lc} u_t=u_{xx}+xu_x & \mbox{in }x\in\mathbb{R},t>0,\\ u(x,0)=g(x), & \mbox{on ...
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19 views

Hyperbolicity balance laws

I am looking to the hyperbolicity of a system of balance laws. Let us consider in general $$\partial_t u + \partial_x (f(u))=g(u)$$ where $f(u)$ is the flux. Have I to consider simply the Jacobian ...
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1answer
19 views

a $p$-Laplacian Dirichlet-problem with finite elements

I would like to solve the $p$-Laplace dirichlet problem \begin{align*} -\Delta_p u & = 1 \quad\text{in } \Omega:=[0,1]^2 \\ u & = 0 \quad\text{on }\partial\Omega \end{align*} for some ...
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37 views

Existence of weak solution to $-\Delta u =0$, $u|_{\Gamma_1} = g$ and $u_{\Gamma_2} = 0$?

Let $\Omega$ be a compact Riemannian manifold with $\partial\Omega = \Gamma_1 \cup \Gamma_2$ a union of disjoint sets. Let $g \in H^s(\Gamma_1)$ and consider $$-\Delta u = 0 \text{ on $\Omega$}$$ ...
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Partial Differential Equations Coefficients

Given the differential equation $ A_{1}U_{xx}+A_{2}U_{yy}+A_{3}U_{xy}+A_{4}U_{x}+A_{5}U_{y}=U $ what does the value of $(A_{3})^{2}-A_{2}A_{1}$ mean whether its equal, less or larger than zero? I ...
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1answer
29 views

Laplace equation and strong maximal principal.

Let $\Omega=\{x\in \mathbb{R}^n:||x||>1\}$ and $u\in C^2(\bar\Omega)$, $\triangle u=0$ in $\Omega$ and suppose that $\lim_{||x||\rightarrow \infty} u(x)=0$. P rove that $$\max_{\bar\Omega} ...
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1answer
32 views

Classification of this nonlinear PDE into elliptic, hyperbolic, etc.

I wanted to know how one would classify a nonlinear PDE into elliptic, hyperbolic or parabolic forms. The particular PDE I would like to know about would be \begin{align} \partial_t u &= ...
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17 views

Domains whose Green functions is explicit or can be approximated explicitly?

The only examples I keep finding are upper half plane (and tilted) and sphere (eg. Evans). Can you suggest some other domains? If not, how about any good books or papers documenting the progress ...