This tag is for "partial differential equations". As opposed to "ordinary differential equations".
83
votes
4answers
2k views
What do modern-day analysts actually do?
In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
35
votes
1answer
964 views
What is the solution to Nash's problem presented in “A Beautiful Mind”?
I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
25
votes
2answers
608 views
PDEs on manifold: what changes from Euclidean case?
I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds.
For example, do things like Poincare's inequality ...
22
votes
1answer
2k views
Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic
what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.
So far, I have ...
22
votes
7answers
911 views
Partial differential equations in “pure mathematics”
One thing I have noticed about PDEs is that they come from Mathematical Physics in general. Almost all the equations I see in Wikipedia follow this pattern. I can't help wondering whether there are ...
16
votes
1answer
292 views
Seeking Fourier series solution on Laplace equation…still looking, am I on track?
Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps.
The problem as given says:
Consider the BVP for $u=u(x,y)$:
...
13
votes
2answers
519 views
Why are harmonic functions called harmonic functions?
Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
13
votes
2answers
372 views
Intuitive explanation of the difference between waves in odd and even dimensions
Motivation: In odd dimensions, solutions to the wave equation: $u_{tt}(x,t)=\nabla u(x,t)$, $u_t(x,0)=0$, $u(x,0)=f(x)$, ($t\geq 0, x\in \mathbb{R}^n$) have the nice property that the value of ...
12
votes
2answers
560 views
PDEs on Manifolds
I am wondering if there is a general coordinate-independent way to define a Partial Differential Equation on a Smooth manifold.
It is definitely true that in each coordinate neighborhood you could ...
11
votes
2answers
364 views
Connections between K-Theory and PDEs?
I've recently spent some time learning (the very basics of) K-theory for $C^*$-algebras and topological K-theory. Actually, my main fields of interest are PDEs and related topics, in particular ...
11
votes
3answers
498 views
What is the purpose of computing the eigenvalue of a PDE problem?
I understand that eigenvalues have their purpose in linear algebra (e.g. iterative methods won't converge unless the modulus of the spectral radius is less than or equal to one). But when I solve ...
11
votes
0answers
616 views
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 ...
10
votes
3answers
481 views
Do discontinuous harmonic functions exist?
A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, ...
10
votes
2answers
87 views
Existence and uniqueness of Stokes flow
What are the solution existence and uniqueness conditions for Stokes' flow?
$$\begin{gathered}
\nabla p = \mu \Delta \vec{u} + \vec{f}
\\
\nabla \cdot \vec{u} = 0
\end{gathered}$$
Maybe you could ...
10
votes
2answers
541 views
classical solutions of PDE with mixed boundary conditions
Nowadays people usually consider PDEs in weak formulations only, so I have a hard time finding statements about the existence of classical solutions of the Poisson equation with mixed ...
10
votes
1answer
210 views
Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?
Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels.
Feynman-Kac formula is also a pde corresponding to a stochastic process ...
10
votes
1answer
174 views
A type of local minimum (2)
Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is a Lipschitz graph. $S \subset \partial \Omega$ is measurable and $H^{n-1}(S)>0$. ...
10
votes
2answers
268 views
Elliptic regularity in Sobolev spaces of negative order
I am having some trouble with Sobolev spaces of negative order. More precisely I am considering the space $W^{-1,p}(\mathbb{R}^2),$ considered as 'the' dual space of $W^{1,q}(\mathbb{R}^2).$
Question ...
10
votes
1answer
178 views
Regularizing effect of the heat equation
Consider the heat equation on $\mathbb{R}_+\times\mathbb{R}^d$
\begin{align*}
\partial_t u -\Delta_x u &= f, \\
u(0,x)&=u_0(x).
\end{align*}
In the case where $u_0\in L^2(\mathbb{R}^d)$ ...
10
votes
1answer
429 views
Understanding a theorem concerning Sobolev spaces
I have two doubts in the proof of the theorem below. If you want the detaIls can be found here.
Theorem A Let $q$ a given real number $q>p$. Let $u \in W^{1,p}$ be a solution to (E2). Assume ...
9
votes
1answer
229 views
Applications of Pseudodifferential Operators
I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
9
votes
2answers
284 views
Mean value property with fixed radius
I will focus on the real line. Let $f$ be a smooth function on $\mathbb{R}$, if
$\forall x\in\mathbb{R}, r>0$,
$$\frac{f(x-r)+f(x+r)}{2}=f(x),$$
we say that f has the spherical mean value property ...
9
votes
2answers
2k views
Energy norm. Why is it called that way?
Let $\Omega$ be an open subset of $\mathbb{R}^n$. The following
$$\lVert u \rVert_{1, 2}^2=\int_{\Omega} \lvert u(x)\rvert^2\, dx + \int_{\Omega} \lvert \nabla u(x)\rvert^2\, dx$$
defines a norm on ...
9
votes
2answers
399 views
viscosity solution vs. weak solution
I am confused between the two. Is one a subset of the other or they are the same/completely different notions? Say I have an euqaiton $u_t=\mathcal{L}u$ for an elliptic operator $\mathcal{L}$ with bad ...
9
votes
3answers
330 views
A harmonic function which is bounded by $\ln(|x|)$ at infinity
I think I can prove that a harmonic function $u$ on $\mathbf{R}^n$ which satisfies $|u(x)|\leqslant C \ln(|x|+1)$ is constant. But what can we say about $u$ when the absolute value sign of $u$ is ...
9
votes
1answer
232 views
Origin of the name 'test functions'
This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
9
votes
1answer
134 views
Does holomorphic a.e. and continuous imply holomorphic everywhere?
Suppose $D$ is a domain in $\mathbb{C}$, $f:D\rightarrow \mathbb{C}$ is a continuous function.
Suppose $f$ is holomorphic outside the zero set $f^{-1}(0)$, and $f^{-1}(0)$ has Lebesgue measure zero.
...
9
votes
2answers
239 views
Reversing the Ricci flow
Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which
inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest
paths on the surface. If ...
9
votes
1answer
727 views
How to compute the first eigenvalue of Laplace operator in an ellipse?
Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$.
It is a fact that the eigenvalue problem for the Laplace ...
9
votes
1answer
87 views
How do we know that PDE solutions obtained via separation of variables are the only ones?
You can find solutions to, for example, the 1D Schrödinger equation $-\frac{\hbar^2}{2m}\Psi_{xx}(x,t) + V(x, t)\Psi(x, t) = i\hbar\Psi_{t}(x,t)$ by assuming solutions of the form $\Psi(x,t) = ...
9
votes
1answer
167 views
Can anyone recommend some books on PDE in $L^p$ space for me?
I need a book covering $L^p$ theory (is it?) on PDE. Stuff should include: De Giorgi-Nash-Moser’s iteration, Harnack inequalities and Schauder estimates on elliptic/parabolic homogeneous/heterogeneous ...
8
votes
4answers
244 views
Roadmap to SPDEs
I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory.
I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and ...
8
votes
2answers
190 views
Are there n-th roots of differential operators?
In analogy to a Dirac operator, it seems to me that formally, the equation
$$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$
is solved by
$$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$
Is there a ...
8
votes
3answers
1k views
The mathematics of music - why sine waves?
Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal.
But what ...
8
votes
2answers
423 views
Relation of Brownian Motion to Helmholtz Equation
one can obtain solutions to the Laplace equation
$$\Delta\psi(x) = 0$$
or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
8
votes
5answers
931 views
Blow up of a solution
What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up.
Thanks.
8
votes
2answers
222 views
Where to learn algebraic analysis
I have been studying categories, sheaf cohomology and complex analysis (the basics since I know just a little). Then recently I tried to find out more about algebraic analysis and these microlocal ...
8
votes
1answer
233 views
Question about a proof in Evans
On page 57. in Partial Differential Equation by Lawrence C. Evans, he prove the maximum principle for the Cauchy problem of the heat equation, i.e. (I quote)
Suppose $u\in C^2_1(\mathbb{R}^n\times ...
8
votes
2answers
330 views
How to argue this consequence?
Suppose that $\Omega=\mathbf{R}^n_+$ and consider a function $0<u<\sup\limits_\Omega u=M<\infty$ such that:
$$\Delta u+u-1=0 \ \ \text{in} \ \ \Omega,$$
$$u=0 \ \ \text{on} \ \ ...
8
votes
1answer
225 views
Nonclassical solution to $u_t-\Delta u=f$ in one space dimension?
I know that one may find continuous $f$ (say in a ball $\bar B$ centered at $0$) such that there exists no classical ($\mathcal C^2$) solution $u$ to the Poisson equation $\Delta u =f$ in $B$, and ...
8
votes
2answers
156 views
Black-Scholes PDE with non-standard boundary conditions
I have the PDE
$$ -\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$
with initial and boundary conditions:
...
8
votes
1answer
152 views
Regularity of elliptic PDE with coefficients in some Sobolev space
Is there a regularity theory for elliptic equations "optimized" for coefficients in a Sobolev space $W^{k,r}$?
By this, I mean a result (and the according elliptic estimates) that gives you (for $k$ ...
8
votes
0answers
247 views
Hille Yosida theorem application
Disclaimer: pretty long and specific (contraction semi groups involved).
I have fourth order parabolic equation
$$
u_t + \Delta^2 u = 0
$$
on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
7
votes
3answers
316 views
Counterintuitive PDE
After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following:
Given the PDE
$\dfrac{\partial f}{\partial x} = 0 \quad $ on $U ...
7
votes
2answers
239 views
Sobolev space is an algebra
How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
7
votes
1answer
243 views
$C^\infty_{0}(\bar\Omega)$ dense in $W^{k,p}(\Omega)$: the closure is necessary?
It is a fundamental result of Sobolve space that
Let $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, then
$C^\infty_{0}(\bar\Omega)$ is dense in $W^{k,p}(\Omega)$.
However, in some literatures, ...
7
votes
1answer
261 views
Textbook that brings together linear algebra and PDEs?
I'm looking for a textbook that goes into as much detail as possible about the parallels between linear algebra in finite, countable, and continuous "spaces." Specific topics that I'm trying to get a ...
7
votes
1answer
209 views
PDE - Feynman-Kac vs. finite difference methods
I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
7
votes
2answers
115 views
Mean Value Property of Harmonic Function on a Square
A friend of mine presented me the following problem a couple days ago:
Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
7
votes
1answer
184 views
Do eigenfunctions of elliptic operator form basis of $H^k(M)$?
We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
