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

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Examples where $1 \in W_0^{k,p}\left( U \right)$

$M$ is a Riemannian manifold, $U$ is a domain in $M$. Consider the Sobolev space $W_0^{k,p}\left( U \right)$: the closure of $C_0^\infty \left( U \right)$ (smooth functions with compact support) in ...
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302 views

Intuition and applications for the p-Laplacian

Consider the p-Laplacian of a suitably nice function $u$: $\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$ Are there useful ways of thinking about the p-Laplace operator, or of thinking about ...
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600 views

What connections are there between number theory and partial differential equations?

What connections are there between number theory and partial differential equations?
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420 views

Weighted Poincare Inequality

I'm trying to prove a result I found in a paper, and I think I'm being a bit silly. The paper claims the following: By the Poincare inequality on the unit square $\Omega \subset \mathbb{R}^2$ we have ...
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1k views

Variational formulation of Robin boundary value problem for Poisson equation in finite element methods

So I am confused about some details of obtaining a variational formulation specifically for Poisson's equation. I am in a Scientific Computing class and we just started discussing FEM for Poisson's ...
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315 views

How should I understand a PDE that contains distribution or measure mathematically?

We know that the Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, $$ \delta(x)=\begin{cases}+\infty, ...
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641 views

Understanding of the Mean Value Theorem in PDE

I learned the following theorem in Folland's Introduction to Partial Differential Equations(p.69 Chapter 2): Suppose $u$ is harmonic on an open set $\Omega\subset{\mathbb R}^n$. If $x\in\Omega$ ...
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1answer
713 views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically in page 6,equation 8 , not equation 9... There ...
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86 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
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190 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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114 views

nonlinear pde equation

I want to solve the problem: find some non-trivial particular solution of nonlinear PDE. Are the any methods for this? I understand that there is no general method to find general solution, but.. One ...
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108 views

Looking for a first order perturbation of the Laplacian having 0 in its spectrum

I would like to find an explicit example of a linear elliptic operator having the following form: $$Lu=-\Delta u +b(x)\cdot \nabla u, $$ where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that ...
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138 views

The existence of harmonic function

If $\Omega$ is a connected bounded open set in $\mathbb R^n$ such that the boundary $\partial \Omega$ is smooth. Then can we find a function $u \in C(\Omega^c)$, such that $\Delta u=0$ in the ...
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1answer
1k views

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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67 views

Natural question about harmonic functions

Let $U$ an open, bounded and convex set in $R^n$. Let $(u_k)_{k \in N}$ a sequence of functions defined in $\overline{U}$. Suposse that each $u_k \in C(\overline{U}) \cap C^{2}(U)$ and harmonic in ...
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132 views

PDE uniqueness by energy method contradicts non-uniqueness???

Consider $$u_t - \Delta u + u = 0$$ $$\frac{\partial u}{\partial \nu} = 0$$ $$u(T) = u(0)$$ on a domain $\Omega$ (the BC is obviously on $\partial\Omega$. If $u$ solves this PDE, clearly as does ...
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152 views

Green Function in $\mathbb{R^3}$ (bounded)

Find the Green function for the domain $Ω = {x = (x_1, x_2, x_3) ∈ \mathbb{R^3} : x_3 ∈ (0, L)}$. The operator in questions is the Laplacian and the Green's function is defined as: $G(x,y) = ϕ(x,y) - ...
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364 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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88 views

Reference for compution adjoint of the operator $L=\Delta^2$

I need to use adjoint operator of the partial differential operator $L=\Delta^2$, where $\Delta$ denotes the Laplacian. I do not want to put this computation in my thesis, because I feel is a bit ...
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133 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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Laplace's equation via Fourier transformation

I have a Laplace equation with some data along the $y$-axis: $$ \begin{cases} u_{xx} + u_{yy} &= 0 \\ u(0,y) &= f(y) \\ u_x(0,y) &= g(y). \end{cases} $$ There is no information of any ...
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581 views

Learning differential/Riemannian geometry for PDEs

I know there have been threads on which books to learn DG/RG from but hopefully this is sufficiently different to avoid closure. Can anyone recommend a book to learn DG/RG (whichever is appropriate) ...
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310 views

KDV PDE: energy constant in time

Show that if u solves the KDV equation $u_t + u_{xxx} + 6uu_x = 0$ for $x \in \mathbb{R}$, $t > 0$ then the energy $\int_{-\infty}^{\infty} \frac{1}{2} u_x ^2 - u^3 \,dx$ is constant in time. ...
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487 views

reference for Navier-Stokes equation

For understanding the Navier-Stokes equations, are there any references which may include one or more of the followings: mathematical rigorousness motivation preliminaries introduction etc.
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Solving the PDE $u_{tt}+2u_{tx}+u_{xx}=2c$

Consider the second-order parabolic inhomogeneous second-order PDE $$ u_{tt}+2u_{tx}+u_{xx}=2c $$ I have seen two ways to solve this problem. I would like to know (1) if Solution 1 is correct (2) if ...
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110 views

The second Friedrichs' inequalities?

In paper On the Validity of Friedrichs' Inequalities,$\Omega$ is a bounded convex domain of $\mathbb{R}^d$, $d=2,3$. Then $$ \tag{1}\qquad \|\mathbf{u}\|_{1,\Omega} \le ...
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80 views

Need source for elliptic regularity on unbounded domains

I need a source that provides a $W^{2,2}$-regularity result for linear elliptic systems on unbounded domains. To be more specific, I study the equation $$ -\Delta w - ic \partial_1 w + \left( ...
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179 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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306 views

Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
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759 views

Weakly Harmonic Functions (Weak Solutions to Laplace's Equation $\Delta u=0$) and Logic of Test Function Techniques.

In analysis we often use test functions $\phi\in C_{0}^{\infty}(U)$ in order to make some kind of deduction about another function $u:U\mapsto\mathbb{C}$. For example, if one can obtain the ...
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199 views

Invertibility of laplacian operator

Let $\Omega\in\mathbb{R}^n$ be a bounded open set with smooth boundary. How to prove the invertibility of $$- \triangle:H^2_0(\Omega) \to L²(\Omega) $$ The injectivity is easy. But how to prove ...
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508 views

How do I solve a PDE with a Dirac Delta function?

I have a PDE in the form of $$ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} + u = \delta(x-1), $$ with initial condition $u(x,0)=100$. I'm trying to solve it numerically, but I have ...
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467 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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261 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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1answer
331 views

Is it true that the Laplace-Beltrami operator on the sphere has compact resolvents?

We consider the Riemannian structure on the sphere $\mathbb{S}^n$ seen as a submanifold of $\mathbb{R}^{n+1}$ and the Laplace-Beltrami operator defined on $C^\infty(\mathbb{S}^n)$ by the equation ...
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384 views

Formulae for PDEs : Commuting derivatives and/or integrals

Many times I come across some new formula being used to work with and/or reduce partial differentials. As kleingordon said, these things are mysteriously not taught anywhere(atleast in physics ...
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411 views

Does Uniform Boundedness in the Sobolev Space $W^{1,2}$ and Convergence in $L^p$ $(1 \leq p < 6)$ Imply Convergence in $L^6$?

Let $B$ denote the open unit ball in $\mathbb{R}^3$. I want to either prove or disprove that a sequence of functions $u_m$ in the Sobolev space $W^{1,2}(B)$ which is uniformly bounded in the ...
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308 views

Partial differential equation & Lax-Milgram Theory?

Consider the elliptic problem $Lu = \exp(x)$ on $[0,1]$ with $Lu = -\frac{d^2u}{dx^2} + \frac{du}{dx}$ and boundary conditions $u(0) = 5$, $\frac{du}{dx}(1) + u(1) = 2$. Answer the following ...
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99 views

My first partial differential equation attempt

have I solved this correctly? My textbook is asking for the relation between $ \alpha $ and $ \beta $: $$ \frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}} $$ Textbook's proposed ...
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245 views

Solutions to the PDE :$2V\frac{\partial I}{\partial V}+2W\frac{\partial I}{\partial W}=I$

While working on engineering problem, I came across this PDE: Let $c_1,c_2$ be two real numbers. Find a continuous function $I:\mathbb{R}_{\geq 0}\times\mathbb{R}_{\geq 0}\rightarrow \mathbb{R}$ such ...
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110 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
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Analytical solution to PDE

I am trying to solve the following linear pde where $u=f(x,y)$ in the domain $y \in (0,\infty)$: $$y\dfrac{\partial{u}}{\partial x} = \dfrac{\partial^2 u}{\partial y^2}$$ with boundary conditions: ...
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195 views

How to prove that the spectrum of the Laplacian over $\Omega\subset \mathbb{R}^n$ is negative?

I am looking for a proof of this well known fact and I guess it has to do with integration by parts (Green's identity). Unfortunately, I only know about 1-d integration by parts( I am just 3rd ...
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145 views

Self adjointness of an elliptic differential operator

Let $A:D(a)\to L^2(\mathbb{R}^n)$ be an elliptic partial differential operator $$ A(f)=\sum_{i,j=1}^{\infty}\partial_{x_j}(a_{ij}(x)\partial_{x_i}f) $$ where $a_{ij}\in C^{\infty}_b(\mathbb{R}^n)$, ...
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267 views

How should a numerical solver treat conserved quantities?

Since the good old Noether Theorem ultimately states that any physical system will exhibit conserved quantities, how should they be treated best in numerical solvers? On the one hand, one can observe ...
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134 views

d'alembert's formula

I'm studying the Cauchy problem for the wave equation $n=2$; $$\begin{cases}u_{tt}=\alpha^{2} u_{xx}, x \in\mathbb{R}, t>0\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} ...
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303 views

Finding the general solution of a quasilinear PDE

This is a homework that I'm having a bit of trouble with: Find a general solution of: $(x^2+3y^2+3u^2)u_x-2xyuu_y+2xu=0~.$ Of course this should be done using the method of characteristics but I'm ...
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498 views

How to solve this Initial boundary value PDE problem? [SOLVED]

Today I came across a question on PDE which makes me really frustrating. The question is to solve this initial boundary value problem using method of separation variables: $$u_{tt}=9u_{xx}\text{ for ...
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107 views

Specific solution to the $1D$ wave equation

So my current solution to the $1D$ wave equation is (with my given boundary and initial conditions): $$y(x,t) = \sum_{n=1}^\infty C_n\cdot \sin\frac{n \pi x}{2 l}\cdot\cos\frac{n \pi c t}{2 l}$$ ...
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371 views

How to use FEM to solve a PDE

I come from a Computer Science background and I've been around searching for a concise tutorial that tells me how Finite Element Method is used to solve a certain partial differential equation. ...