Tagged Questions

2
votes
3answers
57 views

PDEs in biology

I am student who mostly heard lectures on partial differential equations and homogenization. But I really like the idea of working in biology or with biologists - but (with my lack of overview) it ...
1
vote
3answers
87 views

Thermodynamics for math majors

I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis. I'd like to do some reading into the field of thermodynamics. ...
1
vote
1answer
76 views

The choice of the additive constant in the fundamental solution of the Laplace operator

What we usually call "fundamental solution of the Laplace operator" is the following function defined on $\mathbb{R}^n\setminus\{0\}$: $$\tag{1}\Phi(x)=\begin{cases} \frac{-1}{2\pi} \log r & n=2 ...
2
votes
2answers
138 views

What about some engaging PDE topics for undergraduates?

Does anyone have any suitable PDE topics (research or otherwise) for an undergraduate math student? Consider the student has only completed an introductory class in PDE. The student also has ...
2
votes
2answers
153 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
3
votes
2answers
157 views

Good reference texts for introduction to partial differential equation?

As the title, are there any good reference texts for introduction to partial differential equation?
1
vote
0answers
117 views

Intuition for PDE Change of Variables

The algebraic manipulations for changing variables in PDE/ODE problems are often very simple once you know the transformation to use (at least at my level it's just applying the chain rule carefully). ...
4
votes
0answers
109 views

Recommendation textbooks on D-module

I am going to take part in a seminar on D-module and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, and A Primer of Algebraic D-Modules ...
1
vote
0answers
155 views

Subharmonic/Superharmonic Inequality in Gilbarg/Trudinger [Section 2.8]

This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof. Definition: A ...
3
votes
3answers
261 views

What should a PDE/analysis enthusiast know?

What are the cool things someone who likes PDE and functional analysis should know and learn about? What do you think are the fundamentals and the next steps? I was thinking it would be good to know ...
82
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 ...
4
votes
1answer
195 views

Which Fourier transform should I use in PDE?

I learned two "different" Fourier transforms in the real analysis course. For any function $f\in C({\bf R/Z;C})$, and any integer $n\in{\bf Z}$, we define the $n$th Fourier coefficient of $f$, ...
22
votes
7answers
891 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 ...
4
votes
3answers
307 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 $x\in\Omega$ ...
9
votes
1answer
164 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 ...
3
votes
1answer
207 views

How does the boundary property usually work in PDE?

This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of the PDE textbook(e.g. Folland's Introduction to Partial Differential ...
7
votes
2answers
679 views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
7
votes
5answers
895 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.