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

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An inequality used in elliptic PDE

$$\sum a^{ij}\xi_i\eta_j\leq\epsilon \sum a^{ij}\eta_i\eta_j+\frac{1}{4\epsilon}\sum a^{ij}\xi_i\xi_j$$ The summations are $1\leq i,j \leq n$, all the variables are positive. Can anybody prove this? ...
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8 views

$C^\alpha$-regularity of elliptic PDE when $f$ is only continuous

Consider $\Omega\subset\mathbb{R}^n$, open bounded, $$ Lu=f\text{ in }\Omega,\quad u=0\text{ on }{\partial\Omega}, $$ with $Lu=a^{ij}D_{ij}u+b^{i}(x)D_iu+c(x)u=f(x)$, $a^{ij}=a^{ji}$, $L$: strictly ...
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1answer
162 views

Finding fundamental solution to the biharmonic operator $\Delta^2=\Delta(\Delta)$?

Show that when $n=2$, the function $u(x)=-\dfrac{1}{8\pi}\left\lvert x\right\rvert^2\log\left\lvert x\right\rvert$ is a fundamental solution to the biharmonic operator ...
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27 views

Lagrangian (Lagrange multiplier) in Navier-Stokes equations (steady incompresible)

First: Stokes problem (abstract analysis). Let $X$ and $M$ Hilbert spaces and $a:X\times X\longrightarrow\mathbb{R}$ and $b:X\times M\longrightarrow\mathbb{R}$ bilinear and bounded operators. We ...
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41 views

some important proofs about adjoint operators [duplicate]

I was told that the formal adjoint of the gradient is the negative divergence. Let $A : H\to H$ be a bounded, linear operator, The adjoint of $A$, i.e. $A^*: H\to H$ satisfies \begin{equation*} ...
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1answer
15 views

How to perform the following change of variables?

Suppose I have a function $f(t,x,y)$ such that $$f_t = \nabla^26f(t,x,y).$$ I want to perform a change of variables so that for $f(t',x',y')$ we have $$f_{t'} = \nabla^2f(t',x',y').$$ Expanding the ...
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11 views

Uniqueness of solutions for a linear differential equation on a manifold

I have the following situation: $M$ is a smooth manifold. Let $A_t$ be a smooth family of real functions on $M$,that is: $A:I \times M \to \mathbb{R}$ is smooth. (For each $t \in I ,\, A_t \in ...
3
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1answer
173 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
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0answers
18 views

When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

When are $\frac{1}{|x|^s}$ for $s>0$ and $\log|x|$ integrable near the origin? I'm reading Evans PDE and in the construction of the fundamental solution of Poisson's equation, he defines $$ \Phi(x) ...
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1answer
29 views

Solution to the wave equation in $\mathbb{R}^{3}$ with certain initial data

Suppose $f$ is a smooth function satisfying $f(0) = f'(0) = 0$. The question I am working on is to determine the solution $u$ to $u_{tt} - \Delta u = 0$ in $\mathbb{R}^{3}$ with $u(x, 0) = f(|x|)/|x|$ ...
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3answers
121 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
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12 views

Can one explicitely construct a sequence of functions of compact support approximating $u\in W_{0}^{1,p}(\Omega)$?

We define $W^{1,p}_{0}(\Omega)$ as the closure of $C_c^{\infty}(\Omega)$ in the $W^{1,p}$-norm (or equivalently as the closure of the $W^{1,p}$-functions with compact support). Given $u\in ...
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0answers
37 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
2
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1answer
253 views

Weak solution and Rankine-Hugoniot condition

I know from PDE lectures that in case of the following Cauchy problem: $$ \left\{ \begin{array}{l} u_{t}+uu_{x}=0\\ u(x,0)=g(x)\\ \end{array} \right. $$ if a piecewise $C^{1}$ function ...
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1answer
22 views

Separation of variables to get two separate ODEs

[ so far I have: u(x,t) = f(t) h(x). h(x) df(t)/dt+ f(t) d^h(x)/dx^2=- ρ f(t) h(x) 1/f(t) df(t)/dt + 1/h(x) d^2 h(x)/dx^2 =- ρ is my solution so far correct? I'm confused from here on, what do I ...
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36 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
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23 views

Differential Operator simplifying

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
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2answers
45 views

Solving Simple Partial Differential Equation

I don't remember how i can solve this simple partial differential equation. Can someone help me? $$x\frac{\partial \phi}{\partial x}+y\frac{\partial \phi}{\partial y}+ (\alpha+1-x)\phi =0$$ Update ...
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1answer
590 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...
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1answer
22 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
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1answer
65 views

What does the term “regularity” mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...
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291 views

References on the Nash-Moser implicit function theorem

To learn, the Nash-Moser implicit function theorem, I tried the document Hamilton (1982) The Inverse Function Theorem of Nash and Moser, but the article is very encyclopedic. I have a ...
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Boundary conditions for a radiative heat transfer problem

Consider the heat equation $$ \frac{\partial T}{\partial t} - a\Delta T + \mathbf v \cdot \nabla T = S $$ where $S$ is a source term dependent of the radiation intensity $I$ and the temperature $T$. ...
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17 views

Choosing between SOV/Green's functions/Laplace transform for solving PDE - Guideline for choosing the most appropriate method?

Forgive me if this questions seems silly, but I have a question which is keeping me busy. I'm not really looking for a mathematical proof (but it is welcome), however I'm more looking for guided ...
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26 views

How to scale a problem involving heat and the flow of a viscous fluid?

I recently got set this problem and I was wondering if anyone would be able to give me some help on the later parts. An incompressible thermal conducting fluid is contained between two infinite ...
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1answer
25 views

Reflection from the upper half ball to the whole ball is harmonic

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in ...
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1answer
70 views

Solving Bessel's ODE problem with Green's Function

If we have an inhomogeneous boundary value problem $x^2 y'' + xy' + (x^2 -1)y = x,$ $y(0) = y(b) = 0,$ where $b>0$ How to use Green's Funtion to Solve this problem. I am facing issues with ...
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1answer
29 views

Solution of $xu_x + yu_y = 0$

I have the first oder PDE $$xu_x + yu_y = 0 \; \text{on} \; \mathbb{R}^2$$ and I found the solution of that PDE is $$u(x,y) = f\left(\frac{y}{x}\right) = e^C = K$$ which is a constant solution. So, ...
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1answer
25 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
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1answer
23 views

Duals of Sobolev Spaces vanishing on parts of the boundary

I am revising for a Finite Elements course and have the following question about the definition of $H^{-1}$. Let $D\subseteq \mathbb{R}^2$ be bounded Lipschitz domain and let $\Gamma_0 \subseteq ...
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1answer
20 views

Poisson functional on bounded domain

I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below: $$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$ I ...
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2answers
226 views

Can I follow a graduate course in PDE without having studied ODE

Hi I am considering taking the first course on Partial differential equations at my university next semester. I have already taken a first course on functional analysis . I haven't taken a proof based ...
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3answers
312 views

Weakly differentiable but classically nowhere differentiable

Is there any example of a function which is weakly differentiable but none of its versions are classically nowhere differentiable (or differentiable only on a set of measure 0) ? Thanks
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PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
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1answer
41 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
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1answer
21 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
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2answers
63 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
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14 views

PDE Variable transform (integral) Chain Rule

Can somebody help me with the variable transform where the new variable is integral variable? This is a moving boundary problem where the radius of particle, $R(t)$ changes with time. The equations ...
2
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1answer
46 views

solution of 1st order PDE

Find the solution of PDE, $$u_xu_y = u$$ with the initial condition $u(x,0) = 0$ in the domain $x \geq 0$ and $y \geq 0$. I have try the method of characteristic, but it seems like not working for ...
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Existence and uniqueness of Dirichlet problem

Let $U= \{x \in \Bbb R^n: |x|>1 \}$. Suppose $u \in C^2 (U) \cap C(\bar U)$ is a bounded solution of the following Dirichlet problem: $\Delta u=0 \in U$ and $u=\phi$ on $\Gamma=\{x \in \Bbb R^n: ...
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Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
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How do I find out the Partial Differential Equation for the given expression? [on hold]

$$z=x^2f(y)+y^2g(x)$$ Well, I found out that $$ z_x=2xf(y)+y^2g'(x)\\ \ z_y=x^2f'(y)+2yg(x).$$ How do I relate them to get the partial differential equation?
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1answer
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Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
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1answer
76 views

Integration by parts for general measure?

Let $\mu$ be a general measure, suppose $f,g$ has compact support on $\mathbb{R}$, when does the integration by parts formula hold $$\int f'g d\mu = - \int g'fd\mu?$$ I know in general this is false, ...
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Some clarification on nonlinear PDEs

In Strauss' book $\textit{Partial Differential Equations: An Introduction}$, one of the important PDEs listed on p. $2$ is the shock wave equation given by $$u_x + uu_y= 0 \text{.}$$ It is nonlinear, ...
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34 views

Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
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2answers
110 views

Going from a weak formulation to a pointwise a.e statement; don't understand text (PDEs, sobolev spaces)

I just read this: For $u \in H^1(Q)$ where $Q=\cup_{t \in (0,T)}\Omega \times \{t\}$, we have that $$\int_{\Omega}u_tv +\nabla u \cdot \nabla v = \int_{\Omega}fv\quad\text{for all $v \in ...
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0answers
27 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
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1answer
450 views

Relationship between integral equations and partial differential equations

In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem. I've never ...
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2answers
35 views

Difference between two solution of inhomogeneous linear equation

Show that the difference between two solutions of an inhomogeneous linear equation $Lu =g$ with the same $g$ is the solution of the homogenous equation $Lu=0$ I know the definition of linearity, but ...