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

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1answer
358 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 ...
6
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1answer
249 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 ...
6
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1answer
269 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
6
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1answer
83 views

counterexample for ill posedness of the laplace equation

Consider the wave equation with initial data: $$u_{tt}(t,x) + u_{xx}(t,x) = 0$$ $$u(0,x) = u_0(x)$$ $$u_t(0,x) = u_1(x)$$ Hadamard showed that this problem is ill-posed: there exist large solutions ...
6
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1answer
76 views

About some properties of the heat kernel

Maybe my questions are stupit, but I'm so unsure about some properties of the heat equation. Suppose we have a compact and smooth manifold M (possibly with boundary), then in a lot of books they ...
6
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1answer
304 views

Wave Equation Non-uniform string (PDE)

The wave equation in a non-uniform string is : $$ u_{tt} = c(x)^2 u_{xx} $$ $$ u(x,0) = f(x) = e^\frac{(x-\mu)^2}{2 \sigma^2} , \:\:u(0,t) = 0\:,\:\:u(L,t) = 0, \:\: u_{t}(x,0) = -cf'(x) $$ ...
6
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1answer
200 views

How should I numerically solve this PDE?

I am hoping to figure out the function $u(x,y,t)$ for some integer arguments when $u(x,y,0)$ is given (by figuring out I mean generating some images in MatLab), also time $t \ge 0$. $$\frac{\partial ...
6
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1answer
299 views

solving the PDE of a beam under a moving load using Laplace transform

Solve this PDE using Laplace transform : $$ EI {\partial^4 y(x,t)\over\partial x^4}+\mu {\partial^2y(x,t)\over\partial t^2}= F(x,t) $$ $$F(x,t)= P\delta(x-u) / ...
6
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1answer
457 views

Uniqueness for 3-dimensional heat equation initial Robin boundary value problem (SOLVED)

Let $\Omega \subset \mathbb{R}^3$ be a bounded domain. Using an energy argument, show that the IBVP \begin{align} u_t &= \Delta u ~~~~~~~~~~x \in \Omega, ~t>0\\ \frac{\partial u}{\partial \nu} ...
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0answers
114 views

When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar, cylindrical or spherical coordinates ...
6
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1answer
157 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 ...
6
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124 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial ...
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0answers
151 views

Separation of Variables for the Wave Equation

I'm trying to understand the method of separation of variables. I'm probability overlooking something simple, regarding the justification for the term-by-term differentiation that comes up when an ...
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0answers
128 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
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0answers
76 views

How are boundary conditions formally captured by the jet bundle approach to differential equations?

In the jet bundle approach to differential equations https://en.wikipedia.org/wiki/Jet_bundle#Partial_differential_equations one identifies the equation with the set of a solution of the ...
6
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1answer
370 views

weak solutions versus classical solutions

Let $ \Omega $ be an open subset with compact closure of a Riemannian manifold $ M $. Let $ u \in H^1_{0}(\Omega) $ be a weak solution of the Dirichlet boundary problem: $$ -\Delta u + qu = f \; \; ...
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0answers
88 views

Understanding orthogonality in two-scale asymptotic expansion (cf. G. Allaire)

This question is about equation (2.16) of Lecture 2 on Homogenization in Porous Media_ by Allaire page 28. There are two spacial scales: $x$ being macroscopic and $y=\dfrac{x}{\varepsilon}$ being ...
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4answers
287 views

delta functions $e^{x}\delta (x)=\delta (x)$

How would you prove that; $$e^{x} \delta (x)= \delta (x)$$ Is it anything to do with the following relationship; $$ \int_{-\infty}^{\infty} g'(x)h(x)\,dx = \int_{-\infty}^{\infty} g(x)h'(x)\,dx.$$ ...
5
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3answers
2k views

Composition of a harmonic function.

I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help? If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...
5
votes
4answers
488 views

Why separation of variables works in PDEs?

The method of separation of variables is used in many occasions in the upper level physics courses such as QM and EM. But when it is used there is no clear reason why using it is permitted it except ...
5
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3answers
165 views

Why such function does not exist?

I could not prove the following: A function $f \in \mathscr{C}^2([0, \pi])$, such that $$f(0) = f(\pi) = 0,\\ \int_0^{\pi} (f'(x))^2dx = 1,\\ \text{and }\int_0^{\pi} (f(x))^2dx = 2$$ Then such ...
5
votes
2answers
712 views

What is the intuition behind a function being 'weakly differentiable'?

As part of an optimization paper I am reading now, they are talking about a function $g$ being "weakly differentiable". I looked it up on the wiki but I do not have enough context to start cracking ...
5
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2answers
70 views

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 ...
5
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2answers
340 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 ...
5
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1answer
682 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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1answer
2k 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 ...
5
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2answers
108 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 ...
5
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2answers
506 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 ...
5
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1answer
343 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, ...
5
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3answers
760 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 $x\in\Omega$ ...
5
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1answer
208 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, ...
5
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2answers
120 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 ...
5
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2answers
116 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 ...
5
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2answers
153 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 ...
5
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2answers
191 views

How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms?

How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms on a bounded domain? I hear there is a way to do it by RRT but any other way ...
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2answers
78 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 ...
5
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2answers
206 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 ...
5
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2answers
175 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) - ...
5
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1answer
90 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 ...
5
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2answers
149 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 ...
5
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3answers
1k views

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 ...
5
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1answer
316 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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1answer
550 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.
5
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2answers
88 views

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 ...
5
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2answers
342 views

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
5
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1answer
122 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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1answer
93 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( ...
5
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1answer
901 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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1answer
221 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 ...
5
votes
2answers
623 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 ...