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

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Intuitive explanation of Duhamel's principle

This is with regards to the first section of http://en.wikipedia.org/wiki/Duhamel's_principle. I want to see if I am understanding this. Basically the inhomogeneous equation says that heat is being ...
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1answer
277 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
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128 views

How to prove a PDE preserves mass?

My general question is: suppose you are given a PDE (possibly with boundary conditions). What does it mean to say the PDE "conserves mass"? Specifically, if you are given the PDE $$- \nabla \cdot ...
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419 views

Green's function in 2D

How does one compute the Green's function of the laplacian on $ \mathbb{R}^2 $? Can someone point me to a reference? In particular, the fourier transform: $$ \int_0^\infty \int_0^\infty \frac{e^{i m ...
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280 views

continuity and $C^2$ solution of a series

For $\alpha$ with $|\alpha|=2$ let $P$ be a homogenous harmonic Polynom of degree $2$ with $D^\alpha P\ne0$ (e.g. take $P=2x_1x_2$). Choose $\eta\in C^\infty_0(\{x:|x|<2\})$ with $\eta=1$ when ...
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238 views

An elegant non-technical account on the work of Joseph Fourier.

It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
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147 views

How to prove this multivariable function is constant?

Suppose the multivariable function $z=f(x,y)$ is defined on $\mathbb R^2$, has continuous partial derivatives and always satisfies $$x\frac{\partial f}{\partial x}(x,y)+y\frac{\partial f}{\partial ...
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898 views

A PDE exercise from Gilbarg Trudinger. problem 2.2

Prove that if $\Delta u=0$ in $\Omega\subset \mathbb{R}^n$ and $u=\partial u/\partial\nu=0$ on an open smooth portion of $\partial \Omega$, then $u$ is identically zero. I found a proof here, but I'm ...
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907 views

Sobolev embedding for $W^{1,\infty}$?

From the Evans'PDE book I learned $W^{1,\infty}(U)$ coincide with Lipschitz continuous $C^{0,1}(U)$ with $U\in\Bbb{R}^n(n\geqslant1)$ is bounded and $\partial U\in C^1$. I wonder the counterpart ...
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230 views

Can you deduce Neumann boundary data from Dirichlet boundary data?

Say for the following problem, suppose boundary of $\Omega$ is $C^{1,1}$: $$ \left\{ \begin{aligned} -\Delta \phi &= \mathrm{div} \,\vec{u}\quad \text{ in } \Omega \\ \phi&=0 \quad \text{ ...
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957 views

Inhomogeneous Second Order PDE

Given $3u_{tt} + 10u_{xt} + 3u_{xx} = \sin(x+t)$ find the general solution. I have yet to solve any inhomogeneous second order PDE (or even first order ones at that). For homogeneous PDE of same ...
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2answers
175 views

Weird Al Yankovic's Partial Differential Equation

In Weird Al Yankovic's music video "White and Nerdy" (1:20-1:36), there are flashes of a partial differential equation: $$\left(-\frac{h^2}{2\mu}\nabla^2 - \frac{e^2}{r}\right)\psi(r)=E\psi(r)$$ ...
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403 views

An inequality of J. Necas

The above inequality belongs to J. Necas. Sur les normes equivalentes dans $W^k_p$ et sur la coercivite des formellement positives. Sem. math. sup. Universite Montreal, 102-128 (1966). But I can't ...
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113 views

Showing $E(\Omega)$ is a Hilbert Space

Let $E(\Omega)=\{ u\in \{L^2 (\Omega)\}^n : \text{div } u \in L^2(\Omega)\}$, that is $E(\Omega)$ consists of vector valued functions $u=(u^1, \cdots , u^n)$ where each component function $u^i$, $i\in ...
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187 views

Sobolev spaces - about smooth aproximation

Consider $\Omega $ a open and bounded set of $R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow u$ in ...
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308 views

Physical interpretation: weighted eigenvalues of the Laplacian with a potential

I've already posted this question on Physics.SE, but I thougth it could be useful to ask also here. No problem if moderators will ask me to cancel this thread... But, please, have mercy! :-D Let ...
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189 views

Fourier Transform $\displaystyle{F}^{-1}({e^{i\xi^3t}})$

The problem is$$u_t+u_{xxx}=0,u(x,0)=f(x),$$ Use Fourier Transform we get$$\overline{u}=e^{i\xi^3t}\overline{f},$$I want to solve out $u$ . Thus I want to know ...
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392 views

Liouville Theorem

Liouville theorem for superharmonic functions states that Any bounded function $f:\mathbb R^n\to\mathbb R$ admitting an inequality $\Delta f\leq 0$ on $\mathbb R^n$ is a constant function. Here ...
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231 views

Difficulties in solving a PDE problem

This is an exercise in "Variation et optimisation des formes", chapter 3, Ex. 3.8. The preliminaries are: $$D=(0,1)^2,\ f \in L^2(D),\ x_{ij}=(i/n, j/n),\ 0<i,j<n,$$ $$\Omega_n = D\setminus ...
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Finding integrating factor when IF will be a function of x and y

I'm not finding any resource or description or systemic methodology to find integrating factors when the integrating factor will be a function of both x and y. I'm on this problem, $$ ( y - xy^2 ) ...
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192 views

What physical information does the mean value property of heat equation convey?

I'm reading through the Evans' book on PDE, the chapter on heat equation. The definitions are the same as here. I see that mean value property of heat equation is useful for proving maximum principle ...
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342 views

The relationship between two definitions of star-shaped domain

There are two definitions of star-shaped domain. One is given in wikipedia as follows. Def1: A set $S$ in the Euclidean space $\mathbb{R}^n$ is called a star domain (or star-convex set, star-shaped ...
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388 views

Surjectivity of the trace operator in Sobolev spaces

Suppose $U$ is an open bounded set with $C^1$ boundary. It is a well-known result in the theory of Sobolev spaces $W^{1,p}$ that there is a continuous linear operator $T:W^{1,p}(U)\rightarrow ...
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384 views

Regularity of a solution of Laplace equation

Assume $\Omega$ is some open, bounded domain with smooth boundary - say $\Omega = B(0,1) \subset \mathbb{R}^3$. Let $v$ be a solution of the Laplace equation \begin{equation} \begin{cases} \Delta v =0 ...
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87 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 ...
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731 views

An inequality in Evans' PDE

In Section $9.2$ Theorem $5$ of Lawrence Evans' Partial Differential Equations, First Edition the author proves that for a large enough $\lambda$, the equation $$\begin{array}-\Delta u+b(\nabla ...
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655 views

Gradient of a Harmonic Function

I was asked the following vector calculus problem: Let $D$ be the unit ball and let $S$ be the unit sphere in $\mathbb{R}^3$. Suppose that $F:\mathbb{R}^3\rightarrow \mathbb{R}^3$ is a $C^1$ ...
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124 views

Why do we take the odd extension?

When we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x \geq 0 \\ u_t(x,0)=g(x), x \geq 0$$ can we apply Green's theorem or ...
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120 views

Transforming a PDE given basis vectors

I have a non-orthogonal coordinate system defined by $\mathbf x=\mathbf x(r,\beta,z)$, and so I can find the basis vectors as $$ \mathbf g_r=\frac{\partial \mathbf x}{\partial r},\quad\mathbf ...
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69 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
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294 views

Approximate Holder continuous functions by smooth functions

Let $g \in C^{\alpha} (B_1)$ be given. Can we find a sequence $(f_n) \subset C^{\infty} (B_1)$ such that $f_n \rightarrow g$ in $C^{\alpha}(\overline{B_1})$? If so, how can it be done? I have tried ...
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252 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
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111 views

Euler Darboux PDE solution

Consider the Euler-Darboux PDE $$ u_{xy}+\frac{k}{x-y}(u_{x}-u_{y})=0 $$ What is the solution when $k>0$? All text books I have looked at give solutions for $k<0$ and I don't seem to see how I ...
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3answers
548 views

Elliptic estimates on compact manifolds

Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it. For example, for $L$ a linear elliptic operator, (eg. ...
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434 views

What is the Laplace operator's representation in 3-sphere-coordinates?

The three-dimensional Laplace operator in spherical coordinates can be expressed as $$\Delta_3 = \frac1{r^2}\partial_r(r^2\partial_r) + \frac1{r^2} L^2$$ where $L^2$ is the squared angular momentum ...
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254 views

differential operator on manifold

I am currently trying to understand the local expression of a (pseudo)differential operator $$ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $$ on a manifold $M$ (compact and boundaryless, ...
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195 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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144 views

Differential equation - Green's Theorem

I want to find the solution of the following initial value problem: $$u_{tt}(x, t)-u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$ ...
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1answer
96 views

Surveys: problems, conjectures, and questions in some areas of nonlinear analysis

I would like to create a "big-list" of resources (e.g., survey papers, webpages, conference proceedings, monographs, etc.) that collect and offer some context and ...
6
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131 views

Solution of $\frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0$

I've been trying to to solve the following PDE: \begin{equation} \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0 \end{equation} I ...
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140 views

Regularity of the heat kernel

Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous ...
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80 views

Nonlinear PDE $u_y=(u_x)^3$

I need to show that the only solutions of $u_y=(u_x)^3$ that are smooth on whole $\Bbb R^2$ are of the form $ax+by+c$, could anyone help me please?
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131 views

What “standard estimates for the laplacian” do the authors of this paper mean?

I am trying to follow the proof of lemma 2.1 in this paper. The setup. Consider a solution $v$ to the nonlinear equation $$ -\Delta v = ic \partial_1 v + v(1-\vert v\vert^2) ~\mbox{on}~ ...
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1answer
79 views

Solution form for Stokes flows

If $p:\mathbb{R^3} \rightarrow \mathbb{R} $ and $u: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ satisfy: $$\nabla p-\nabla^2u=0$$ $$\nabla\cdot u=0$$ How can we prove that every solution is of the form: ...
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354 views

Wave Equation Non-uniform string (PDE)

The wave equation in a non-uniform string is : $$ u_{tt} = c(x)^2 u_{xx} $$ $$ u(x,0) = f(x) = e^\frac{(x-\mu)^2}{2 \sigma^2} , \:\:u(0,t) = 0\:,\:\:u(L,t) = 0, \:\: u_{t}(x,0) = -cf'(x) $$ ...
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184 views

Caccioppoli-Leray Inequality for De Giorgi's regularity theorem

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
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223 views

How we do actually compute the topological index in Atiyah-Singer?

I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing ...
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2answers
257 views

Trace regularity result $\lVert n \times u\rVert_{H^{-1/2}}$

There is a result in a paper I am reading : Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that $$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
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1answer
390 views

Parabolic linear PDE with continuous coefficients; how to solve and explanation of text needed

I have the following PDE $$u_t = a(x,t)u_{xx} + b(x,t)u_x + c(x,t)u - f(x,t)$$ $$u|_{t=0} = u_0$$ over the domain $S^1 \times [0,T)$. The coefficients and $f$ are in $C^{k,\alpha}$ for some $k$ (and ...
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287 views

Capacity theory beginner resources

I'm currently studying a book on shape optimization: Variation et optimisation de formes: Une analyse géométrique By Antoine Henrot, Michel Pierre. The book introduces at some point capacity, and uses ...