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

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Bounding an integral on boundary of Lipschitz domain

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain with bounded boundary $\Gamma.$ So $\Gamma$ is a hypersurface of dimension $(n-1)$. I want to show that $$\int_\Gamma ...
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54 views

Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \begin{equation} \phi(R)\equiv ...
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180 views

Derive Poisson integral formula in a ball

Trying to derive by myself the Poisson integral formula in a unit ball. I should get $$\Delta u=0 \,\text{ in } B(0,1), \,\,\, u(x)=\varphi(x)\,\,\text{at } \partial B(0,1) \Longrightarrow \\$$$$u(x) ...
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43 views

line of sources for differential equation

It is well known that for representation of point source with some intensity $q(t)$ in some PDE we use delta-function $\delta({\bf r} - {\bf r}')$. If problem requires usage of line of sources which ...
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66 views

Can a wave vanish at two times?

Suppose $u$ is a solution of $$ \begin{cases}u_{tt} = \Delta u & \textrm{ in } \mathbb R \times \mathbb R^n, \\ u(0,\cdot) = 0, \\u_t(0,\cdot) = g(x), \end{cases} $$ with $g$ compactly ...
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71 views

How prove this $f(x,y)=\dfrac{1}{2\pi}\ln{\dfrac{1}{|x-y|}}+\dfrac{i}{4}-\dfrac{1}{2\pi}\ln{\dfrac{k}{2}}-\dfrac{C}{2\pi}$

let $$J_{0}(x)=\sum_{p=0}^{\infty}\dfrac{(-1)^p}{(p!)^2}\left(\dfrac{x}{2}\right)^{2p}$$ and ...
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62 views

Show that $b(\ ., \ .)$ it is not coercive.

Let $b(\ .,\ .)$ a bilinear operator so that $$b(u, \phi)= \int_0^T((u(t), \ \phi(t)))dt- \int_0^T(u(t), \ \phi'(t))dt,$$ where $((\ ., \ .))$ it is the inner product of $H_0^1(\Omega)$ (Sobolev ...
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141 views

Difference quotient (pde)

Let $u: U\subset\mathbb{R}^n\rightarrow\mathbb{R}$. The Difference quotient of $u$ is defined by $D_k^hu(x)=\dfrac{u(x+he_k)-u(x)}{h}$ with $h\in\mathbb{R}$, $0<|h|<\textrm{dist}(V,\partial ...
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373 views

Reflection principle for harmonic functions

$U^+ \colon= \left\{x\in \mathbb{R}^n\mid |x| < 1, x_n>0\right\}$ is an open half-ball. Assume $u \in C^2 (\overline{U^+}$) is harmonic in $U^+$ with $u=0$ on $\partial U^+ \cap \{x_n=0\}$. ...
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Please explain. Really I dont understand and I need to learn. Pde: : example of finding particular integral

When we look at the solution part, there is a statement The PI of the given PDE is obtained as follows After the statement, I dont really understand all of the calculation. Espacially, After the ...
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68 views

Fundamental solution and dissipation result for biharmonic Heat equation

I guess this is easy a very easy question for some people. References would be appreciate : What is the fondamental solution of the biharmonic heat equation ? and how fast is it decreasing in time ...
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55 views

Is there a typo on this definition?

This is from Iishi's User's guide to viscosity solution. I don't understand the $\ni$ in the definition 2.6 at the end of its first line, is it a typo? User's guide page 11
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129 views

How to find the eigenvalue and eigenfunction of Laplacian?

Define a bounded domain $\Omega=(0,a)\times(0,b)$ What is the eigenvalue and eigenfunction of the Laplacian with homogeneous boundary condition? my first thought is something like $sin(n\pi ...
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57 views

Consequence of strong maximum principle

Let $\Omega \subset \mathbb{R}^N$ be a bounded, connected, open, and regular set. Let $u \in C^\infty(\Omega)$, such that $$u=0, \mbox{ on }\partial \Omega.$$ Let us suppose that as a consequence of ...
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242 views

Geometry: Finding a surface that is orthogonal to a given family of surfaces

Problem Consider a one-parameter family of surfaces ${S_c}$ in $\mathbb{R^3}$ (with $c$ being a real constant) described by $$S_c = \{(x, y, z)|f(x, y, z) = c\}, c ∈ \mathbb{R}$$ (a) Assume that we ...
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112 views

I don't understand this PDE solution involving Fourier coefficients and orthonormal eigenfunctions.

$u_{xx} + u_{yy} = 0$ with $x \in (0,\pi)$ and $y \in (0, \pi)$ Initial Conditions: $$ u(x,0) = x^2 $$ $$ u(x,\pi) = 0 $$ Boundary conditions: $$ u_{x}(0,y) = 0 = u_{x}(\pi, y) $$ I performed ...
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52 views

Series of complex Fourier coefficients.

I've been trying to figure this out for days now, but I have no idea how to show this. It's from Partial Differential Equations: An Introduction by Walter A. Strauss. Suppose $\int_{-\pi}^{\pi} [ ...
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175 views

Verifying a solution from Strauss' “Partial Differential Equations - An Introduction”, 2nd edition

In section 9.2 on page 241, question #12 is given as follows: "Solve the three-dimensional wave equation in $\{r\ne0,t>0\}$ with zero initial conditions and with the limiting condition ...
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42 views

Separating Partial Differential Eq

I have a PDE: $$ \frac{\partial^2\phi(r,\theta)}{\partial r^2} + \frac{1}{r}\frac{\partial\phi(r,\theta)}{\partial r} + \frac{1}{r^2}\frac{\partial^2\phi(r,\theta)}{\partial\theta^2} + ...
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106 views

Partial Differential Equation $u_t+u_x=\cos(c-t)$

Given $u_t+u_x=\cos(c-t)$ where $u(x,0)=\dfrac{1}{1+x^2}$ . Find the solution $u(x,t)$ using characteristic method. I have found $\dfrac{dt}{ds}=1$ and $t(0)=0\implies t=s$ $\dfrac{dx}{ds}=1$ and ...
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318 views

2D Laplace equation with mixed boundary conditions on the upper half-plane.

Find the temperature distribution $T(x,y)$ in the upper half-plane, given that the temperature along the $x$-axis is at: $$T(x,0)=T_0, \quad x<-1$$ $$T(x,0)=T_1, \quad x>1$$ And ...
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45 views

Proprieties of the heat kernel on a bounded domain of $\mathbb{R}^{d}$

I'm interesting to PDE, and I'm asking if the heat kernel with Dirichlet boundary conditions $p_{D}(t,x,y)$ on $[0,1]^{d}$, where $d\geq 1$ satifies i) $\int_{D}p_{D}(t,x,y)dy =1$ or $c_{0} ...
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77 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
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52 views

problem with the phase portrait of a equation

I have to analyze the phase portrait of this equation: $\dot{x}=y \\\dot{y}=x+\alpha y-x^{3}+6\alpha xy^{2}$ I´ve already checked when $\alpha=0$, i.e $\dot{x}=y \\\dot{y}=x-x^{3}$ i got a saddle ...
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91 views

Solve the wave equation explicitly on a helf-line with b.c.

A question from my pde homework: Let $\alpha$ be constant, $\alpha \neq -1.$ Consider the wave equation on $x>0$, $t>0$ with the following data: $$u_{tt}-u_{xx}=0 \;\;\;\;\text{for $x>0$, ...
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About infinitesimals and differentiating in high dimensions.

I got a little confused here. Please take a look and help me to express correctly the notation below. Let $f:\Re^n \rightarrow \Re$ be a real valued function over $n$-dimensional space of column ...
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68 views

Prove the estimate $|u(x,t)|\le Ce^{-\gamma t}$

Assume that $\Omega \subset \Bbb R^n$ is an open bounded set with smooth boundary, and $u$ is a smooth solution of \begin{cases} u_t - \Delta u +cu = 0 & \text{in } \Omega \times (0, \infty), \\ ...
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37 views

The term “maximal solution” for PDE

A solution $x(t)$ of the ODE is called maximal if it is defined on an open interval and cannot be extended to any larger open interval. from "Ordinary Differential Equation". Alexander ...
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198 views

On max-min representation for the principal eigenvalue of second order elliptic operator

(Just to be upfront about things, this is a homework problem.) I'm asked to show that the principal eigenvalue, $\lambda_1$ of an uniformly elliptic operator can be represented by \begin{equation} ...
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57 views

FETI domain decomposition - kernel of local stiffness matrices

Consider the differential equation \begin{align*} -\Delta u&=f\mathrm{\ in\ }\Omega \\ u & = 0 \mathrm{\ on\ }\partial\Omega \end{align*} with $\Omega=(0,1)^2$. We're splitting $\Omega$ into ...
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145 views

Lyapunov Stability

Let $\dot{x}=v(x)$ with $v(x)=Ax+O(\left \| x \right \|^2)$, $v\in C^k(U)$, $U\subset \mathbb{R}^n$, $ n\geq 2$, is true or not that if the origin is a singular point Lyapunov stable for ...
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563 views

How to write Pfaffian differential equation

I am studying the example. And I dont understand how to write the pfaffian diff equation in the first line. What is its formula?
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Maximum principle homework

Let U=$R_{+}^2$, $u \in C^2(U) \cap C(\overline U)$ with $\Delta u=0$ in U. If in addition, u is bounded above on U, prove that: $sup_{U} u=sup_{\partial U} u$. we can apply the maximum principle to ...
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402 views

Find the integral curves of the equation

Question: Find the integral curves of the equation: $$\frac{dx}{y^2x-2x^4}=\frac{dy}{2y^4-x^3y}=\frac{dz}{2z(x^3-y^3)}$$ I could not find any similar example to understand this type of questions ...
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372 views

how to prove mean value property for harmonic functions?

For a harmonic function $u(x)$, on domain $\Omega$ where $y \in \Omega \subset \Bbb R^n $, how to show that $$ u(x) = \frac{1}{\omega_n R^{n-1}}\int_{\partial B_R(y)} u(\sigma) d\sigma$$ where ...
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115 views

On how to get a solution for a nonhomogeneous problem for the heat equation.

Evans PDE book presents the following problem (page 87): Write down an explicit formula for a solution of $$\left\{\begin{matrix} u_t-\Delta u+cu=f &\text{ in }\mathbb{R}^n\times(0,\infty) ...
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192 views

Diffusion-advection equation with time-variable coefficients

Is the fundamental solution (Green's function) of the 1D advection-diffusion equation $$\frac{\partial{\phi}}{\partial{t}} = D(t)\frac{\partial{^{2}\phi}}{\partial{x^{2}}} - ...
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229 views

Fourth order differential equation

I have this physics mathematical problem : (see link in comment) $$EI \frac{∂^4u}{∂x^4}= f \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ The boundary conditions are: ...
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159 views

What really Navier-Stokes existence smoothness problem is?

Can any one explain to me (without using mathematical equations) that what is Navier-Stokes existence smoothness problem. I read a lot about Navier Stokes existence smoothness problem, but I still can ...
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351 views

The Decision of three methods of the solutions $dx/P=dy/Q=dz/R$

Question: (A) $$\frac{adx}{(b-c)yz}= \frac{bdy}{(c-a)xz}=\frac{cdz}{(a-b)xy}$$ (B) $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{1-z^2}$$ These are simultaneous diff eq. of the first order and the ...
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Existence and uniqueness of solution for an eliptic problem.

Let $\Omega \subset R^n$ be an open and bounded set with smooth boundary and $f \in C^2(\Omega)\cap C(\overline{\Omega})$. Let $ a \geq 0$ be a constant. Consider the following problem: $$ ...
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Derivation of weak form of Euler Lagrange Equation

In Giaquinta's and Giusti's 1982 paper entitled "On the regularity of the minima of variational integrals", they look at the following quadratic functional: \begin{equation} ...
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348 views

steady state solution to differential equation - checking my work

EDIT: fixed a stray negative sign. The problem as given: $y'' + 2y' + 5y = 10\cos t$ We want to find the general solution and the steady-state solution. We're using $\mu y'' + c y' + k y = F(t)$ ...
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58 views

Existence of solution in Hölder spaces

Let's say we have a PDE, for example the Laplace equation: $$ \Delta u = f. $$ Usually, to solve such a thing, one finds its variational formulation, and solves it in some Sobolev space. My question ...
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54 views

What exactly is a multi-scale operator and what does it do?

In a paper I'm reading at the moment we're concerned with a third order nonlinear ODE for which we know the solution near thr origin look something like an upside-down parabola crossing the y axis at ...
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112 views

Solve the initial value problem (PDE)

$$2u_t + x(1+t)u_x = u^2$$ $$u(x,0)=x$$ Tried to do this by method of characteristics. So, using change of variables as follows $$\left\lbrace \begin{array}{c} p=x \\ q = 2\ln(x)-(t+t^2/2) ...
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57 views

How to find $u(x,y)$?

I have the second-order PDE $6u_{xx}+u_{xy}-u_{yy}=0$ . The change of variables I'm given is $s=x+2y$ and $j=x-y$ . I used the chain rule and solved $u_{xx}$ and $u_{xy}$ and $u_{yy}$ in terms of ...
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272 views

Finite difference for variable conductivity

I'm trying to discretize a portion of the heat equation for a sphere and for a cylinder where: $r$ = radius, $T$ = temperature, and $k$ = thermal conductivity. for the cylinder shape: ...
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392 views

Heat equation with Neumann boundary condition

Background: In our PDE class we explored the heat equation with Dirichlet boundary condition $$u_t - \Delta u = 0 \;\text{ in } \Omega \subset \mathbb{R}^n \;\text{bounded}\\ u = u_0(x) \;\,\text{at} ...
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70 views

Why is the case $1/2<\rho\leq 1$ trivial in proving the following inequality?

I'm studying Elliptic Partial Differential Equations by Q. Han and F. Lin. In Lemma 1.41 is given the elliptic equation $D_j(a_{ij}D_i u)=0$ where the coefficient matrix $(a_{ij})$ is constant ...