# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

178 questions

**32**

votes

**1**answer

3k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**8**

votes

**2**answers

773 views

### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**12**

votes

**1**answer

768 views

### When is a given matrix of two forms a curvature form?

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of 1-forms $...

**5**

votes

**1**answer

342 views

### Fredholm index vs. Limit cycle theory

Let $A$ be the algebra of all smooth functions $f: \mathbb{R}^2 \to \mathbb{R}$ such that $f$ is flat at the origin and is real analytic on $\mathbb{R}^2 \setminus \{0\}$.
Let $B $ be ...

**99**

votes

**16**answers

10k views

### Does Physics need non-analytic smooth functions?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ...

**58**

votes

**7**answers

11k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....

**37**

votes

**1**answer

4k views

### Unconditional nonexistence for the heat equation with rapidly growing data?

Consider the initial value problem
$$ \partial_t u = \partial_{xx} u$$
$$ u(0,x) = u_0(x)$$
for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: [...

**7**

votes

**2**answers

626 views

### Willmore minimizers for genus $\geq 2$

For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as
$$
\cal W(f) = \int _{\Sigma} \frac{1}{4} |\vec H|^2 d \mu_g,
$$
where $\vec H$ is the mean ...

**5**

votes

**1**answer

772 views

### propagation of singularities & the Schrodinger equation

I've been thinking about the following propagation of singularities result:
Let $X$ be a compact manifold, and let $P$ be a differential operator (of, say, order $m$) on $X$ whose principal symbol $\...

**6**

votes

**1**answer

336 views

### Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$

How is the proof that
$$[L^2(0,T;X)]' = L^2(0,T;X')$$
looking like, where $X$ is a Hilbert space? I am asking for the proof that the dual space of $L^2(0,T;X)$ is the space $L^2(0,T;X^*)$.
Is the ...

**46**

votes

**6**answers

6k views

### Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation
I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...

**16**

votes

**5**answers

2k views

### “Physical” construction of nonconstant meromorphic functions on compact Riemann surfaces?

Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic ...

**27**

votes

**1**answer

2k views

### The Riemann zeros and the heat equation

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
2\sum_{...

**10**

votes

**1**answer

2k views

### Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact?
in which book we have this fact, the number of ...

**9**

votes

**3**answers

2k views

### Reference request: Simple facts about vector-valued Sobolev space

Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...

**9**

votes

**1**answer

1k views

### Density of smooth functions in Sobolev spaces on manifolds

Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have ...

**9**

votes

**1**answer

383 views

### Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...

**15**

votes

**3**answers

1k views

### Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...

**9**

votes

**2**answers

1k views

### Characterize where the Dirichlet Problem for the Laplacian is always solvable

Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...

**5**

votes

**2**answers

1k views

### Maximum principle for weak solutions

Hello,
maximum principles for parabolic PDEs seem to be well-known, if the solution is a priori C^2 (cf. Protter, Weinberger: Maximum principles in differential equations). However, what about weak ...

**3**

votes

**1**answer

1k views

### Method of characteristics of a system of first order pdes

I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first ...

**10**

votes

**3**answers

413 views

### Reference request: Systems of linear PDES with constant coefficients

I am looking for a reference for the following statement:
Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs
\begin{align}
P_i(\partial / \partial x_1, \dots, \...

**3**

votes

**1**answer

181 views

### Elliptic problem on a domain split in two subdomains

Consider the following elliptic problem in a split domain:
$$ (\ast) \quad\begin{cases} -\Delta u=f_1 \quad &\text{ in } U_1\\
-\Delta u =f_2 & \text{ in }
U_2\\
u=g & \text{ on } \...

**7**

votes

**2**answers

651 views

### Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...

**7**

votes

**2**answers

2k views

### Estimates on the Green function of an elliptic second order differential operator.

Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free ...

**7**

votes

**3**answers

376 views

### Books and resources on PDEs that use Mathematica and Matlab

Can you recommend some reference books that use software like MATLAB and Mathematica to deal with the basic topics in
analysis of PDE (the ones you can find in Strauss' book Partial Differential ...

**6**

votes

**2**answers

606 views

### Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold.
Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator?
Thanks

**4**

votes

**1**answer

405 views

### Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of ...

**6**

votes

**2**answers

1k views

### The adjoint operators as elliptic operators

Edit:
It seems that the link "https://cms.math.ca/Events/Toulouse2004/abs/ss7.html#lt" which contains a talk by Loic Teyssier about homological equations and vanishing cycles is temporally ...

**3**

votes

**2**answers

866 views

### System of linear first order PDE with constant coefficients

recently in my researches I've come across the following operator
$$L\left(\begin{array}{c}
a_1\\
\vdots\\
a_n
\end{array}\right)=M_1\left(\begin{array}{c}
...

**3**

votes

**2**answers

558 views

### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}...

**2**

votes

**1**answer

450 views

### Existence of a solution to an infinite dimensional Stratonovich SDE

Let
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
$U_0:=Q^{1/2}U$
$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...

**2**

votes

**1**answer

80 views

### Boundary condition for elliptic problems and domain decomposition

This question is motivated by one that has been previously asked on this website: Elliptic problem on a domain split in two subdomains
Consider an open domain $U$ split in two non-overlapping ...

**1**

vote

**1**answer

528 views

### Properties of the trace term in the Itō formula

Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where
$U,H$ are separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ is nonnegative ...

**4**

votes

**2**answers

964 views

### Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form
$$\cos\left(t\right)\partial_{x}\mathbf{u}+\sin\left(t\right)\partial_{y}\mathbf{u}+...

**1**

vote

**1**answer

195 views

### The study of dynamics of a polynomial vector field via Green's function methods

In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is ...

**62**

votes

**9**answers

16k views

### Why can't there be a general theory of nonlinear PDE?

Lawrence Evans wrote in discussing the work of Lions fils that
there is in truth no central core
theory of nonlinear partial
differential equations, nor can there
be. The sources of partial
...

**65**

votes

**13**answers

6k views

### Counterexamples in PDE

Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their ...

**47**

votes

**9**answers

6k views

### Motivation for and history of pseudo-differential operators

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...

**25**

votes

**1**answer

4k views

### Jet bundles and partial differential operators

A geometric way of looking at differential equations
In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-...

**32**

votes

**4**answers

3k views

### Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...

**22**

votes

**8**answers

6k views

### Applications of PDE in mathematical subjects other than geometry & topology

Partial differential equations have been used to establish fundamental results in mathematics such as the uniformization theorem, Hodge-deRham theory, the Nash embedding theorem, the Calabi-Yau ...

**21**

votes

**5**answers

3k views

### Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...

**24**

votes

**18**answers

3k views

### PDEs as a tool in other domains in mathematics

According to the large number of paper cited in MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural ...

**20**

votes

**7**answers

2k views

### Applications of Microlocal Analysis?

What examples are there of striking applications of the ideas of Microlocal Analysis?
Ideally i'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...

**18**

votes

**2**answers

4k views

### Compact Embeddings of Sobolev Spaces: A Counterexample Showing The Rellich-Kondrachov Theorem Is Sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...

**23**

votes

**2**answers

2k views

### Simulating Turing machines with {O,P}DEs.

Qiaochu Yuan in his answer to this question recalls a blog post (specifically, comment 16 therein) by Terry Tao:
For instance, one cannot hope to find an algorithm to determine the existence of ...

**13**

votes

**2**answers

4k views

### what's the idea behind Carleman estimate

A standard Carleman-type estimate is of the form
$$
\sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty}
$$
...

**29**

votes

**5**answers

2k views

### How to define a differential form on a fractal?

It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one ...

**17**

votes

**1**answer

2k views

### Image of the trace operator

It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...