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

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6
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1answer
111 views

Transforming a PDE given basis vectors

I have a non-orthogonal coordinate system defined by $\mathbf x=\mathbf x(r,\beta,z)$, and so I can find the basis vectors as $$ \mathbf g_r=\frac{\partial \mathbf x}{\partial r},\quad\mathbf ...
6
votes
1answer
250 views

Solving the 2D Poisson equation with variable boundary location

I am trying to find $z(r,\phi)$ from the 2D Poisson equation in polar coordinates: $$\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial z}{\partial ...
6
votes
1answer
68 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
6
votes
1answer
268 views

Approximate Holder continuous functions by smooth functions

Let $g \in C^{\alpha} (B_1)$ be given. Can we find a sequence $(f_n) \subset C^{\infty} (B_1)$ such that $f_n \rightarrow g$ in $C^{\alpha}(\overline{B_1})$? If so, how can it be done? I have tried ...
6
votes
1answer
225 views

Is there an integral form of Newton's method?

Warning : This seems like a silly sort of question, not the kind I'd ask out loud. The contraction mapping theorem is a basic tool for proving existence of, and finding solutions to, equations. Given ...
6
votes
1answer
107 views

Euler Darboux PDE solution

Consider the Euler-Darboux PDE $$ u_{xy}+\frac{k}{x-y}(u_{x}-u_{y})=0 $$ What is the solution when $k>0$? All text books I have looked at give solutions for $k<0$ and I don't seem to see how I ...
6
votes
3answers
486 views

Elliptic estimates on compact manifolds

Hey where may I find elliptic estimates for PDEs on compact (no boundary) Riemannian manifolds? I want a source/paper/book where I can cite it. For example, for $L$ a linear elliptic operator, (eg. ...
6
votes
1answer
381 views

What is the Laplace operator's representation in 3-sphere-coordinates?

The three-dimensional Laplace operator in spherical coordinates can be expressed as $$\Delta_3 = \frac1{r^2}\partial_r(r^2\partial_r) + \frac1{r^2} L^2$$ where $L^2$ is the squared angular momentum ...
6
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1answer
235 views

differential operator on manifold

I am currently trying to understand the local expression of a (pseudo)differential operator $$ \int_{R^n} e^{(x - y)\cdot \xi} \sigma(x,\xi) \, d \xi $$ on a manifold $M$ (compact and boundaryless, ...
6
votes
1answer
162 views

An interesting semi-linear PDE problem

Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi linear equation $$ -\Delta u+c(u)=f\;\;\text{in}\;\mathbb{R^n} $$ where $f\in ...
6
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1answer
84 views

Surveys: problems, conjectures, and questions in some areas of nonlinear analysis

I would like to create a "big-list" of resources (e.g., survey papers, webpages, conference proceedings, monographs, etc.) that collect and offer some context and ...
6
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1answer
109 views

Solution of $\frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0$

I've been trying to to solve the following PDE: \begin{equation} \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0 \end{equation} I ...
6
votes
2answers
136 views

Regularity of the heat kernel

Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous ...
6
votes
1answer
77 views

Nonlinear PDE $u_y=(u_x)^3$

I need to show that the only solutions of $u_y=(u_x)^3$ that are smooth on whole $\Bbb R^2$ are of the form $ax+by+c$, could anyone help me please?
6
votes
1answer
128 views

What “standard estimates for the laplacian” do the authors of this paper mean?

I am trying to follow the proof of lemma 2.1 in this paper. The setup. Consider a solution $v$ to the nonlinear equation $$ -\Delta v = ic \partial_1 v + v(1-\vert v\vert^2) ~\mbox{on}~ ...
6
votes
1answer
79 views

Solution form for Stokes flows

If $p:\mathbb{R^3} \rightarrow \mathbb{R} $ and $u: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ satisfy: $$\nabla p-\nabla^2u=0$$ $$\nabla\cdot u=0$$ How can we prove that every solution is of the form: ...
6
votes
1answer
334 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
votes
1answer
221 views

Characteristics in the theory of PDE's - What's Going On?

What's really going on as regards characterstics in the theory of PDE's? (Hopefully you wont have to check the links provided below, included for those who are interested really) For second order ...
6
votes
1answer
171 views

Caccioppoli-Leray Inequality for De Giorgi's regularity theorem

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
6
votes
1answer
256 views

Why are weak solutions to PDEs good enough?

Obviously solutions in $C^k$ are nicer but it seems people are happy with obtaining weak solutions in some Sobolev space that only satisfy the weak formulation. Why should this, in the real world, be ...
6
votes
1answer
201 views

How we do actually compute the topological index in Atiyah-Singer?

I am taking a lectured class in Atiyah-Singer this semester. While the class is moving on really slowly (we just covered how to use Atiyah-Singer to prove Gauss-Bonnet, and introducing ...
6
votes
2answers
248 views

Trace regularity result $\lVert n \times u\rVert_{H^{-1/2}}$

There is a result in a paper I am reading : Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that $$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
6
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1answer
375 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
263 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
296 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
92 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
votes
1answer
80 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
votes
1answer
211 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
324 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
votes
2answers
259 views

Weakly differentiable but classically nowhere differentiable

Is there any example of a function which is weakly differentiable but none of its versions are classically nowhere differentiable (or differentiable only on a set of measure 0) ? Thanks
6
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1answer
570 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} ...
6
votes
1answer
414 views

is $\nabla \cdot ( c^2 \nabla)$ a Laplace-Beltrami operator?

I asked this on mathoverflow, and was suggested to ask it here. Someone mentioned, in passing, to me that $u \mapsto \nabla \cdot ( c^2 \nabla u)$ is a Laplace-Beltrami operator. Does anyone have ...
6
votes
2answers
207 views

Prerequisits for Gauss-Green theorem

Consider the following theorem from the appendix C from Evans PDE book: I know about integration in $\mathbb{R}^n$ but not about how to make sense of the integrals on the right-hand side. As my ...
6
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0answers
131 views

Uniqueness of solutions to $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$

The problem I am working on is to show that there is a unique compactly supported solution to the PDE $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$, $(x, t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0)= ...
6
votes
1answer
131 views

oblique derivative smoothness of harmonic functions

Let $Q$ be a domain in the half-space $\mathbb R^n\cap\{x_n>0\}$ and part of its boundary is a domain $S$ on the hyperplane $x_n=0$. Let $u\in C(\bar Q)\cap C^2( Q)$ satisfy $\Delta u=0$ in $Q$ and ...
6
votes
0answers
71 views

Existence of classical solution for equation $\Delta u + v(x) u = 0$

It is known that if $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$ then the Dirichlet problem $$ \begin{array}{rl} \Delta u & = 0 \quad \text{in $D$}, ...
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0answers
118 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
171 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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0answers
134 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 ...
6
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0answers
163 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 ...
6
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0answers
178 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
81 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
votes
1answer
443 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 \; \; ...
6
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0answers
93 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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0answers
1k views

First Order PDE Solution Method Issues

I'd really appreciate help with two little questions relating to first order partial differential equations. Just to quickly let you know what I'm asking, the first is about solution methods to first ...
6
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0answers
185 views

Show that, given spherically symmetric initial data, a solution to the heat equation is spherically symmetric

Let $\phi \colon \mathbb{R}^n \to \mathbb{R}$ be continuous with compact support. Furthermore, suppose that $\phi$ is spherically symmetric. That is, suppose that $\phi(Tx) = \phi(x)$ for every ...
6
votes
1answer
156 views

Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...
5
votes
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
307 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
votes
2answers
841 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 ...