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

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1answer
333 views

Question about the proof of Harnack's inequality in Evans and Gilbarg's PDE book

I have some trouble in understanding the proof of Harnack's inequality. Since I have consulted two books, I explained my three questions one by one. In Evans' book Partial Differential Equations, 2nd ...
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1answer
55 views

PDE solving using change of variables

I have to solve these PDE using change of variables. 1)$p(x) $$\partial f \over \partial x $$+q(y) $$\partial f \over \partial y $=0 A hint is given as to use $u=\int {1\over p}dx$, $v=\int {1\over ...
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0answers
93 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
2
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1answer
193 views

Clarifying definition of outward unit normal

I would like to figure out how to properly define the outward unit normal vector $\nu$ for a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$ ($n \ge 2$). I am ...
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1answer
22 views

characteristic equation in pde

In the PDE: $ yU_y-xU_x=1$ how did the characteristics become $dx\over -x$=$dy \over y$ =$du \over 1$.Can someone please expalin how these charactristic equations were obtained
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1answer
20 views

Elliptic operator with real coefficients on $\mathbb{R}^2$?

Supposedly, an elliptic first order differential operator on $\mathbb{R}^2$ with real coefficients does not exist. But $$p_m(\xi_1,-a_1 \xi_1/ a_2)=-a_1 i \xi_1 +a_1 i /xi_1=0$$ for any real $\xi_1$, ...
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0answers
32 views

Finding the homomorphism that links the linear part of a dynamical system to the nonlinear part.

here is a picture of my problem Basically what i have is that i was told i could find this homomorphism by doing the following ...
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1answer
119 views

numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
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1answer
46 views

weak form of the problem in two domains

Let $\Omega$ be an open, bounded domain, and a smooth internal boundary $\Gamma$ divides $\Omega$ into two open and connected sets, $\Omega1$ and $\Omega2$, where $\Omega1$ is strictly included in ...
2
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2answers
169 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
1
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1answer
24 views

On the proof of a $W^{2,p}$ estimate - regularity on eliptic PDE

I see this proof studying the paper http://www.math.uiowa.edu/~lwang/cccalderon.pdf and I couldn't understand what he did. If $||f||_{L^p(B_{4})} = \delta$ is small and the measure $|\{ x \in B_1; ...
3
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1answer
168 views

Behavior of solutions to the heat equation at infinity

I have read that for the solution $u$ of the heat equation $$u_t = u_{xx},$$ with $u(x,0)= a \exp(-bx^2)$ for some $a,b >0$, it holds $$\lim_{x \to \infty} u_x(x,t) = 0 = \lim_{x \to \infty} ...
2
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0answers
124 views

Solving a system of integral-partial differential equations

Hi I am a student in electrical engineering. Currently I am facing a difficult problem solving a coupled integral-differential partial equations arising from mean field game. The problem is similar as ...
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0answers
60 views

Maximum Principle for subsolutions of heat equation with drift term

This question is about a proof in the article Mean Curvature Evolution of Entire Graphs by Klaus Ecker and Gerhard Huisken, The Annals of Mathematics, 2nd Ser., Vol. 130 (page 455). You can find the ...
2
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1answer
92 views

Dual of $div$ on spaces where the tangential value is fixed

Say $\Omega$ is a domain in $\mathbb R^3$ with a smooth boundary $\Gamma$. Consider the spaces $$ H_{n,0}=\{v\in H^1(\Omega):n\cdot v \bigr |_{\Gamma} = 0\} $$ and $$ H_{t,0}=\{v\in ...
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0answers
42 views

Can you explain this partial derivative approximation?

How do they go from the left side to the approximation on the right? What does $j$ mean in this summation? $$ {\partial f_i^n \over \partial t} \approx \sum_j {\partial f_i^n \over \partial \hat x_j} ...
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1answer
47 views

Question on Gagliardo-Nirenberg.

On page 679 of this paper, the authors claim they can get a special case of Gagliardo-Nirenberg with a constant of 1/2. They prove this using functions in $C_0^\infty(\mathbb{R}^2)$, for which the ...
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1answer
251 views

2d laplace equation with neumman boundary condition

$$\Delta u(x,y)=0$$ $$x,y\in(0,1),$$ $$\frac{\partial u(0,y)}{\partial x}=0,\quad \frac{\partial u(1,y)}{\partial x}=0,\quad\frac{\partial u(x,0)}{\partial y}=0,\quad\frac{\partial u(x,1)}{\partial ...
2
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1answer
69 views

Is the solution of the initial-boundary value problem correct?

I have to solve the problem: $$u_t=u_{xx}-6x, 0<x<L, t>0$$ $$u(0,t)=0, u(L,t)=2L^3, t>0$$ $$u(x,0)=x^3, 0<x<L$$ I did the following: $$u(x,t)=v(x,t)+s(x)$$ So we have the following ...
3
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0answers
76 views

Which came first, energy minimization or pde?

I'm interested in a historical perspective on pde. I would like to know more about the original derivation of pde. It seems like d'Alembert was working on the one dimensional wave equation $$ ...
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1answer
40 views

PDE initial conditions

I have a pde: $y_{xx}-y_{tt}=4$. By using the substitution $v=x-t, u=x+t$ I have boiled it down to $y(x,t)=a(x+t)+b(x-t)+x^2-t^2$ however I have initial conditions $y_t(x,0)=0$ and $y(x,0)=sin(x)$. ...
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0answers
33 views

Consequence of elliptic estimates up to the boundary

Consider the half space $\Omega=\{x=(x_1,\ldots,x_N)\in\mathbb{R}^N:x_N>0\}$. Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ a positive bounded solution of $$ \begin{eqnarray*} \Delta ...
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0answers
58 views

What is the name of such equation in fluid dynamics? How does it come out?

I read papers and see this equation. Honestly, I am not familiar with fluid dynamics. So I was wondering if this equation is common in fluid dynamics, or it is the special case in this paper. Could ...
4
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1answer
115 views

Elliptic PDEs in Banach space

The standard textbooks discuss weak solutions and regularities in Hilbert spaces $W^{k,2}.$ I could not find a good reference on the theory based on Banach spaces $W^{k,p}.$ It would be good to point ...
2
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1answer
446 views

Method of characteristics for a system of pdes

I can do parts a) and b) as follows $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1\end{pmatrix}\frac{\partial}{\partial{}x}\begin{pmatrix} u \\ v \\ w\end{pmatrix}+\begin{pmatrix} ...
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1answer
55 views

Exists $C$ constant: $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$

Show that there exists $C$ constatant such that $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$. This is a question in my ...
1
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1answer
260 views

Numerical solution of non-linear differential equation with MATLAB

I need some information to know if I can solve a nonlinear integral equation with terms $ u_{x} $ , $ u_{x}.u_{y} $ , $ u_{xx} $ , $ u_{xy} $ $u_{yy} $ $ u_{x}^{2} $ $ u_{y} ^{2} $ By numerical ...
27
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6answers
2k views

Should I understand a theorem's proof before using the theorem?

I find myself embarrassed when using results in books. For example, there are so many results in Sobolev spaces that I think I would not be able to understand all of them. Yes, I could try to ...
2
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2answers
183 views

Greens function of 1-d forced wave equation

[ORIGINAL PROBLEM] You are given hat the Green's function $g(x,t,\xi, \phi)$ is $\frac{\partial^2g}{\partial t^2} - \frac{\partial^2g}{\partial x^2}=\delta(t-\tau)\delta(x-\xi)$ with ...
5
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1answer
334 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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1answer
57 views

Introducing constraint $\int udx =0$ to variational formulation

Consider a Neumann-Poisson problem. \begin{align*} -\Delta u & = f \ \mathrm{in}\ \Omega\\ \frac{\partial u}{\partial\nu} & = g \ \mathrm{on}\ \partial\Omega \end{align*} Then $u$ is ...
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1answer
30 views

Inequality in $L^2$ based norms

Recently I came across the following inequality. Let $f:\,\mathbb{R}^n\rightarrow\mathbb{R}$ be smooth, then the inequality $$\left\lVert f\right\lVert_{\dot H^2(\mathbb{R}^n)}\leq\left\lVert\Delta ...
2
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1answer
45 views

the differentiability from an oscillation estimate

If we have an oscillation estimate of $u\in W^{2,n} (B^{+}) \cap C^0(\bar B^{+})$, and $u=0$ on $T = B \cap \partial R^{n}_{+}$, that $$osc_{B^{+}} \frac{u}{x_{n}} \leq C.$$ Then how comes that $u$ is ...
2
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0answers
38 views

Greens function of a uniformly charged sphere

The potential $\phi(\boldsymbol{x})$ satisfies $\nabla^2\phi=f$ It may be shown that by defining an appropriate Green's function $g(\boldsymbol{x},\boldsymbol{\xi})$ that ...
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1answer
45 views

For Green's function of $\Delta-c$, how to show $\int_{X}G(x,y)(\Delta-c)f(y) dy=\pm f(x)$?

Let $X$ be a compact Riemnnian manifold and $\Delta$ the Laplacian. Suppose that $G(x,y)$ be the Green's function of the elliptic operator $\Delta-c$ for a positive constant $c$. I think the ...
2
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1answer
61 views

Domain decomposition

Could any body help me to understand that how \begin{align*} -\Delta u=f \hspace{0.2cm}\text{in}\hspace{0.2cm}\Omega\\ u=0\hspace{0.2cm}\text{on}\hspace{0.2cm}\Gamma \end{align*} equivalent to ...
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1answer
81 views

Show this inequality in the “heat equation” problem.

Let $(u,t)$ the $C^2$ solution of the equation $$ u_t=u_{xx}+u, \textrm{ over } [0,a]\times[0,T]\subset \mathbb{R}^2 $$ where $T>0$ Show that $$ \max\limits_{[0,a]\times[0,T]} |u| ...
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1answer
75 views

show a PDE has no distribution solution in $\mathbb{R^2}$\{0}

" Consider the following equation in the plane $x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=f(x^2+y^2)$ where $f(t)$ is a $C^\infty$ function of the real variable $t$ such that ...
2
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2answers
170 views

Why are Dirichlet eigenfunctions real-analytic?

Consider the eigenvalue problem $\Delta u + \lambda u = 0$ on some bounded domain $\Omega \subset \Bbb R^d$ with smooth boundary, with Dirichlet data $u|_{\partial\Omega} =0$. It is known that ...
0
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1answer
222 views

Is wave equation a hyperbolic, parabolic, or elliptic PDE?

I am very beginner of PDEs. I want to study wave equation in 1D and 2D for numerical methods. Basic question is which type is a wave equation is, elliptic, parabolic, or hyperbolic?
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28 views

Integrability of second derivative of infinity harmonic functions

Consider the infinity harmonic functions, i.e. solutions of the equation $$ \Delta_\infty u = \langle Du, D^2 u \, Du \rangle = 0. $$ It is known that the solutions are everywhere differentiable ...
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1answer
59 views

Burgers equation with $a(x) = u^2$

I am trying to solve the characteristic equations of $u_t + u^2u_x = 0$ without initial conditions in order to show graphically how profiles get smoothed or develop a shock depending on initial data. ...
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1answer
24 views

Cauchy problem for partial derivative equation

This my first encounter with PDE. Given the equation $x D_x(z)+(y-xz)D_y(z)=z$, where $z=z(x,y)$ and initial conditions $y-x=2z,zx=-1$. The question is: how to interpret these conditions. I have one ...
0
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1answer
59 views

Maximum principle type problem

Suppose $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ is sufficiently smooth and is such that $(1/2) u_{xx} + u_{xy} + 2u_{yy} = 0$ in a ball $B$ centered at the origin. Must $u$ attain a maximum inside ...
0
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1answer
128 views

Connection Between Fundamental Solution and Symmetries of PDE

The typical derivation of the fundamental solution of Laplace's equation is to look for a radially symmetric solution because the Laplace equation has radial symmetry, and a similar heuristic can be ...
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0answers
231 views

How to balance learning and researching as a new PhD student?

As a new PhD student, how to balance learning and researching? I am in Australia and here we don't have any course in PhD period. I know I need to learn something about my programme, but sometimes ...
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1answer
43 views

Existence or nonexistence of semilinear Poisson equation?

Suppose that $\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$. Consider the following semilinear Poisson equation with prescribed Dirichlet data on the boundary: ...
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1answer
42 views

Diffuse equation-type PDE: Help me!

$ d, r, K, l, L $ with $ l<L $ are positive constants. Question 1: Please solve this PDE for function $ I $: $$ \frac{\partial I}{\partial t}=d\frac{\partial^{2}I}{\partial ...
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1answer
42 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 ...
6
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1answer
196 views

Measure theory questions applied to Second Order PDE

Most of the questions are more measure theory and integration related but I need to give some context, so I will now. Consider the quasilinear 2nd-order partial differential equation ...