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

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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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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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1answer
19 views

Problem on energy of a Discrete Galerkin Method

I'm reading an article from this website: article question is in page 3,about a wave equation,and use the Galerkin method to discrete the space. (1) page4 why the author use the fraction ...
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24 views

Elliptical Coordinates Problem

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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1answer
23 views

3D Cauchy Problem

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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2answers
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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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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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1answer
39 views

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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2answers
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Non-linear partial differential equation

I would like to find out if there is any specific method -apart from numerics- for finding solutions of a non-linear PDE of the form $$\nabla \times \mathbf{A} = \pm\lambda\mathbf{A} \tag{1}$$ under ...
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89 views

Partial Differential equations and applications- Reference request

I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below. Partial differential equations: Conservation laws, ...
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2answers
37 views

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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31 views

Is my function singular at these two points?

My function $S(x,y,t)$ satisfies the following PDE $$\frac{\partial S(x,y,t)}{\partial t}=-H\left(x,\frac{\partial S(x,y,t)}{\partial x}\right)$$ where the known function $H$ is singular when ...
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1answer
20 views

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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1answer
45 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 ...
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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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25 views

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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1answer
37 views

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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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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2answers
23 views

Fourier Transform of Poisson Equation

While trying to solve the Poisson Equation by using Green's Function I have to Fourier transform the equation i.e $$-\nabla^{2}\phi(r)=\rho(r).$$ In the book after Fourier transform, the solution ...
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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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1answer
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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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2answers
77 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
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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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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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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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28 views

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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1answer
63 views

Solving a master equation with linear coefficients

I have the following PDE: $$ \partial_t P(x,y,t)=x\partial_xP(x,y,t)+(y-1)\partial_yP(x,y,t)+2P(x,y,t). $$ Mathematica suggests that the solution is $$ ...
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1answer
109 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponenent allows for convenient ...
2
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1answer
45 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
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Relation between the Eigenvalue and the potentials.

Let us consider the following problem , suppose for $i=1,2$ \begin{equation}\begin{aligned} -\nabla \gamma_i \nabla u_i = \lambda^1_i u_i \quad\mbox{in }\Omega\\ u_i =0 \quad\mbox{on }\partial\Omega ...
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1answer
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Solve initial value problem (C.S.I.R)?

The initial value problem is $$ \frac{\partial u}{\partial t} +x\frac{\partial u}{\partial x} = x, \ \ 0 \leq x \leq 1, \ \ t > 0 \ \ and$$ $$ u(x,0) = 2x \ \ has$$ a unique solution $u(x,t) \ ...
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1answer
56 views
+50

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
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existence and uniqueness of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
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1answer
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About Sobolev-Poincare inequality on compact manifolds

Let $2^* = \frac{2n}{n-2}$ where $n$ is the dimension of a compact closed manifold $M$. We get from the Sobolev/Poincare inequality the identity $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u ...
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1answer
430 views

Book recommendations for self-study at the level of 3rd-4th year undergraduate

I have only recently discovered an interested in mathematics and I could only take a year off work to be back at school. Needless to say, for financial reasons (couple of mortgages) I will need to ...
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1answer
30 views

Morrey's inequality

From PDE Evans, 2nd edition, page 281: Now \begin{align} \int_0^s \int_{\partial B(0,1)} |Du(x+tw)| \, dS(w) dt &=\int_0^s \int_{\partial B(x,t)} \frac{|Du(y)|}{t^{n-1}} \, dS(y) dt \\ ...
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viscous burgers equation physical meaning

The viscous Burgers' equation : $$ q_{t} + qq_{x} = vq_{xx}, \:\:\: \mbox{where} \:\:\: v > 0, $$ combines the nonlinear propagation of $q(x,t)$ and the diffusion. What is this equation for? (in ...
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Estimates of positive solutions of $\Delta u=u$ in $\mathbb{R}^n$, $n>2$.

Let $u$ be a positive solution to the problem $\Delta u=u$ in $\mathbb{R}^n$, prove that there are positive constants $C, C_1,C_2$ such that $$ Ce^{-C_1|x|}<u(x)<Ce^{C_2|x|}. $$ Hints are ...
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1answer
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1st order Cauchy problem

Discuss the solution of $u_t+ uu_x= 0$ with the following Cauchy data: a) $u(x,0)= x ; 0\leq x\leq 1$ b)$u(x,0)=1/2; 0\leq x\leq 1$ Sketch the domains in the $x-t $ plane where the solutions are ...
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Regularity of semilinear heat equation

I'm facing following regularity issue and i wonder if anyone of you guys is able to help me. I'd like to show that the solution of a semilinear heat equation is classical, i.e. $C^2$ in space and ...
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Is it necessary to know all the details in proofs of theorems you study in PDE's?

I've been studying PDE from the book by L.Evans for some time now. I came across some statements in the proofs which I couldn't justify. But to complete the exercises I didn't need to know all these ...
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Limit/Integration in heat equation

While studying heat equation from PDE by L.Evans, I came across the following limit which I'm not able to prove. For $n>=1, \delta >0$ , $lim_{t \to 0+} \;\;{1 \over ...
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1answer
35 views

Electromagnetic fields and Laplace equations along a square

I'd like to solve Laplace equation satisfying the following BCs: $$\phi(x,y=0)=0$$ $$\phi(x=0,y)=0$$ $$\phi(x,y=1)=9\sin(2\pi x)+3x$$ $$\phi(x=1,y)=10\sin(\pi y)+3x$$ where $0\leq x,y\leq 1$. I have ...
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intuitive fact of a class of functions defined in $R^n$

I am reading an article and i have the following situation: Let $u: R^n \rightarrow R$ a continuous function in $R^n$. Supoose that u is nonnegative and that for all $t \geq 0$ the set $L_t = \{ x ...
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3answers
49 views

How to solve $(\partial_x + i\partial_y)u - c(\partial_x+i\partial_y)au=0$?

The following equation, $$(\partial_x + i\partial_y)u - c(\partial_x+i\partial_y)au=0$$ ($a=a(x,y)$ and $\partial_x=\frac{\partial}{\partial x}$) with solution, $$u=\exp(ca)f(x+iy)$$ where $f$ and ...
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1answer
25 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
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Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
2
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2answers
56 views

Separation of variables: when to have exponential solution and when sinusoidal?

In separation of variables, one can assume a solution of V(x,y) = X(x)Y(y) and after plugging this into Laplace's equation which is: ${{\partial^2 V} \over {\partial x^2}}$ + ${{\partial^2 V} \over ...
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How to find the solution in $x$ when the differential equation is in $\partial_y$ such that $\partial_y = \partial_x + f(x)$

For my problem, the differential equation can be simplified by defining a new operator like $\partial_y = \partial_x + f(x)$. But at the end I need a solution in terms $x$. How can I do that? As an ...