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

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Equating the general operator to the Laplacian: $Lu=-\Delta u$

This is from Chapter 6 -- Second-Order Elliptic Equations -- of PDE Evans, 2nd edition. See the very bottom of my post for my question. Pages 311-312: $\quad$We will in this chapter study the ...
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show the convergence of the integral $\quad \quad \quad F(t,x)=\int_{-\infty}^{+\infty}\exp[i\tau t-(i\tau)^{1/2}x - (i\tau)^\theta] \,d\tau$

The original problem is : Let $\theta$ be a number such that $1/2<\theta<1$. Prove that $\quad \quad \quad F(t,x)=\int_{-\infty}^{+\infty}\exp[i\tau t-(i\tau)^{1/2}x - (i\tau)^\theta] ...
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About the image of the trace operator for Sobolev spaces.

Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain. Once every convex function is locally Lipschitz, we have that $\partial\Omega$ is Lipschitz, therefore, the trace operator $T: ...
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64 views

Formulas for Schrödinger unitary groups of operators

Let $\Omega$ an open set of $\mathbb{R}^n$. Consider the Hilbert space $X=L^{2}\left(\Omega\right)$ and the Schrödinger operator $A=i\Delta$ defined on the domain $D(A)=H^2(\Omega)$. Is there any ...
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Elliptic regularity on the torus: reference request

Suppose we work on the two dimensional torus $\mathbb T^2$. Let $L_a^2$ be the space of square integrable functions with zero space average and $H_a^m$ be the corresponding Sobolev space. Suppose we ...
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Holder Gradient Estimate for Linear Equation

In Gilberg and Trudinger's book, Elliptic Partial Differential Equations of Second Order, Theorem 12.4 states the following : Let $\Omega \subset \mathbb{R}^2$, and let $u$ be a bounded $C^2(\Omega)$ ...
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Prove the maximum principle for a solution of the PDE $(-\Delta + \lambda)u=0$

Show that if $u$ satisfies $(-\Delta + \lambda)u=0$ in a smooth region $\Omega$ where $\lambda$ is a nonnegative real number, then u satisfies the maximum principle. We can just assume u is smooth, ...
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Green's function for Helmholtz/Schrödinger operator

Does someone have a reference for a rigorous treatment of the Green's function for Helmholtz/Schrödinger operators? In particular under which conditions ($g$, Domain etc.) the application of the usual ...
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Global bounded solution of $u_{tt}=\Delta u-mu+h$ in the Hilbert space $X=H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$

Let $\Omega$ be an open subset of $\mathbb{R^n}$. Consider the linear wave equation $$\begin{cases} \dfrac{\partial^{2}}{\partial t^{2}}u\left(t,x\right)=\Delta ...
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About the comparison of the gradient of two specific harmonic functions on the boundary

Consider $D_2 \supset D_1$, two open, bounded and convex sets with $D_2 \supset \overline{D_1}$. Suppose that exists $r_0 > 0$ such that for each $x \in \partial D_1$ there exists $z_x \in D_1$ ...
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Show that for all $(\tau, \xi) \in \mathbb R^{n+1}$ we have $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$

Show that, for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$ This is the exercise 7.4 in the book by Francois Treves. It is just a fundamental ...
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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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Studying Hamiltonian PDEs - Where to start?

I first got in contact with Hamiltonian PDEs when I wrote my first thesis about the Nash-Moser-Theorem as some books mentioned them as a field of application of this theorem. As those examples were ...
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1answer
73 views

Method of Characteristics for a non-linear PDE

I've been trying to work through some of the more difficult questions we've been given in class in regards to the method of characteristics for solving PDEs, but I've come a bit unstuck. I've been ...
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Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
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Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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How to solve a first order partial differential equation?

I have a first order pde ($P_{14}$ is the unknown function in $a, b, a_1, b_1$): \begin{align} & -\frac{a}{a_1} \frac{\partial P_{14}}{\partial a} - \frac{a b}{a_1 b_1} \frac{\partial ...
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Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
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1answer
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Comparison of the gradients of two harmonic functions near the boundary

Let $\Omega$ an open bounded domain in $R^n$. Let $u,v$ be nonconstant smooth functions in the interior of $\Omega$ and harmonic in $\Omega$. Suppose that $u,v \in C(\overline{\Omega})$ and $u \geq ...
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Regularity for a parabolic problem with nonsmooth coefficients

I'm looking for references on the regularity of the (weak) solution to the parabolic problem with nonsmooth coefficients. In most literature, like Evans, the coefficients are often assumed to be ...
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How to solve a system of first-order partial differential equations?

I have a system of first-order partial differential equations. \begin{align} & -\frac{a}{a_1} \frac{\partial P_{12}}{\partial a} - \frac{a b}{a_1 b_1} \frac{\partial P_{12}}{\partial b} + ...
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Reference for p-capacitary functions

Consider $\Omega_1 $ and $\Omega_2$ two open bounded and convex sets in $R^n$with $\Omega_1 \supset \overline{\Omega}_2$. The unique weak solution of the problem $$ \begin{cases} \Delta_p u = 0 ...
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Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
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PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
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1answer
35 views

Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
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Green's function of a Markov Chain, and maybe of a Feller Process?

How are the Green's functions of a Markov chain related to the notion from PDE theory? For instance, if the Markov chain (i.e. discrete state space) is continuous time, then the Green's function I'm ...
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29 views

About the Hopf lemma for the p-laplacean

My question is about this article :http://arxiv.org/pdf/1204.6578v1.pdf . Consider $U$ an open bounded domain in $R^n$ and $u \in W^{1,p}(U)$ a p-harmonic function in $U$(the definition is on the ...
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Di Perna-Lions theory

I'm reading the paper of Di Perna and Lions "Ordinary differential equations, transport theory and Sobolev spaces". I'm not understanding the proof of corollary II.2; in particular I don't understand ...
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238 views

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from ...
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Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
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Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...
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2answers
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3D Cauchy problem for the PDE $ yu_x-xu_y+u_z=0 $

I will answer the question myself but let me know what you think of my correctness. We have the Cauchy Problem $$ yu_x-xu_y+u_z=0 $$ with data $u(x,y,0) = x+y$.
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1answer
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Using the method of characteristics to find a general solution to PDE

I want to find the general solution to $3U_x-4U_y = x^2$ using the method of characteristics. I'm given the answer which is $U(x,y)=\frac{x^3}{9}+F(3y+4x)$ but I'm having trouble getting to this ...
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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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Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
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1answer
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Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
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Missing explanation in this paper of Masmoudi.

In this paper, on page 4, beginning in the line above 3.8, the authors begin a discussion of a given variational problem. I follow their argument until they begin the line of reasoning that begins ...
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Showing $u = \phi_\epsilon * u,$ ie Function Equals Convolution with Function

My question is the following: Starting with the mean value property for harmonic $u \in C^2(\Bbb R^3)$, ie $$u(x) = {1 \over {4\pi R^2}} \int_{\partial B_R(x)}u(y) dy,$$ deduce that if $\phi \in ...
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Blow up of a solution of the p-Laplace equation at a boundary point

My doubt is about a proof from the lemma 2.8 of the article 18 of this homepage: http://www.math.kth.se/~henriksh/Henriks_page/publications.html The result that I said is: Lemma: Let $D_1$ and $D_2$ ...
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Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
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1answer
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Riesz measure associated with a subharmonic function

In page 101, corollary 4.3.3., from Armitage and Gardiner's book on potential theory, the authors prove that any subharmonic function, can be identified with a positive measure (Riesz measure). In ...
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Definition of elliptic pde

http://en.wikipedia.org/wiki/Elliptic_partial_differential_equation Very confusing. So are there non-linear elliptic pdes? It seems to be the case. And what about $$A u_{xx} + C u_{yy} + F u = 0$$ ...
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Non stationary solutions of the PDE $u_t + u_x = u_{xx}$

Problem. Consider the PDE $$ u_t + u_x = u_{xx}, \qquad (t,x) \in (0,+\infty) \times (0,1). $$ (i) Write the unique solution $\overline{u}=\overline{u}(x)$ which does not depend on time and ...
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Converting from mixed boundary condition to dirichlet boundary condition

I have found a finite element method code in MATLAB that I want to use however the boundary conditions are dirichlet -- more specifically the function is 0 at the boundary. However, the PDE I'm ...
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28 views

Show that Fourier series arising in solution of differential eqn. converges uniformly

Let $f \in L_2(0,\pi)$ have the Fourier expansion $f(x) = \sum_{n=2}^{\infty} f_n\sin(nx)$. Compute (formally) the boundardy value problem $$ u''(x) + u(x) = f(x) \qquad \mbox{ for } 0 < x < ...
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Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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Accuracy of a finite-difference method for numerically solve a PDE or BVP

When solving the Poisson Equation $$-u''(x)=f(x)$$ with Dirichlet-Neuman boundary conditions $$u(0)=0, u'(1)=0$$ using a finite difference 2-order centered scheme and a 2-order upwind ...
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TT* Duality argument

I am trying to obtain $L^p$ estimate for a system of nonlinear PDE. I do not have $L^2$ to work with for my problem, I heard $TT^*$ argument is useful tool if one wishes to mapp from $L^p$ to $L^p$. ...
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1answer
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Why orthgonality matters for numerical treatment of PDEs?

It seems to me that general orthogonal coordinates are quite popular in numerical treatments of PDEs. Sometimes people even use conformal maps to generate the locally orthogonal grids. But the actual ...
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61 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the Green's function. I have to solve the given differential equation using Green's function method $$\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...