# 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.

280
questions

**36**

votes

**1**answer

4k 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. ...

**11**

votes

**2**answers

1k views

### Elliptic operators corresponds to non vanishing vector fields

Added, June 19, 2019: The main motivation of this post is to associate an index to differential operator associated to a dynamical system such that the index has an interesting ...

**45**

votes

**1**answer

5k 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: [...

**14**

votes

**1**answer

1k 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 $...

**4**

votes

**1**answer

628 views

### Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)

Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...

**17**

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 ...

**4**

votes

**1**answer

425 views

### Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...

**5**

votes

**1**answer

380 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 ...

**2**

votes

**0**answers

97 views

### Role of absolute continuity of divergence of BV function in proof of renormalization property

In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
Heuristically, ...

**117**

votes

**15**answers

13k 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), ...

**69**

votes

**7**answers

15k 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....

**23**

votes

**2**answers

7k 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 ...

**30**

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_{...

**7**

votes

**2**answers

837 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

947 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 $\...

**5**

votes

**1**answer

528 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

**1**answer

372 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 ...

**2**

votes

**1**answer

255 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 } \...

**2**

votes

**1**answer

126 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 ...

**7**

votes

**1**answer

280 views

### Lavrentiev phenomenon between $C^1$ and Lipschitz

Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int_a^b f(t,y(t),y'(t))\,dt
$$
such that
$$
\inf_{y\in Lip([a,b])}F(y)<\inf_{...

**4**

votes

**1**answer

228 views

### Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...

**2**

votes

**1**answer

126 views

### Surveys/monographs on the vortex filament equation

Where can I find surveys on the mathematical aspects of the vortex filament equation?
In particular, I'm interested in the following topics:
physical motivation;
notion of solutions and ...

**0**

votes

**1**answer

80 views

### Relationship between the vortex filament equation and the cubic Schrödinger equation

How is the vortex filament equation
$$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$
where $\chi(t,s)$ is a curve in $\mathbb R^3$,
related to the cubic Schrödinger equation?
Note 1. ...

**50**

votes

**6**answers

7k 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 ...

**33**

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{\...

**36**

votes

**5**answers

4k views

### Explicit eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...

**25**

votes

**7**answers

3k 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/...

**11**

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 ...

**12**

votes

**1**answer

2k 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 ...

**12**

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}(\...

**13**

votes

**3**answers

739 views

### A conformal map whose Jacobian vanishes at a point is constant?

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$.
Assume $d \ge 3$ ...

**18**

votes

**3**answers

2k 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

**1**answer

434 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 ...

**7**

votes

**1**answer

2k views

### Solutions to the eikonal equation

Theorem. Let V be a $C^\infty$ function on a riemannian manifold $M$ and $p$ be a nondegenerate local minimum with $V(p)=0$. Then there is a unique positive function $\varphi \in C^\infty(U)$ such ...

**10**

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 ...

**8**

votes

**3**answers

981 views

### Newlander-Nirenberg for surfaces

Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest possible case, namely ...

**6**

votes

**2**answers

2k views

### Maximum principle for weak solutions

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

2k 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 ...

**8**

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

**2**answers

1k 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

**3**answers

410 views

### How to find the associated conservation law from a given symmetry

It is a very well-known fact that any conservation law associated with some given PDE has an associated invariance (by Noether's Theorem). However, it is completely mysterious for me how to compute/...

**10**

votes

**3**answers

539 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, \...

**8**

votes

**1**answer

812 views

### Traces of Sobolev spaces

Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...

**4**

votes

**2**answers

477 views

### Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs

Where can I find a (readable and self-contained) proof of the following result?
Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...

**3**

votes

**2**answers

516 views

### A function in $W^{1,p}(\Omega)$ for $1 < p < n$ which is not differentiable a.e

I'm seeking a function which belongs to $W^{1,p}(\Omega)$ for $p < n$ which is not differentiable a.e. There is a standard theorem which shows that if $p > n$ then in fact any function in $W^{1,...

**7**

votes

**3**answers

638 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

**0**answers

277 views

### A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...

**6**

votes

**2**answers

899 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

325 views

### A Liouville theorem involving an advection term

I am curious whether there are any Liouville theorems for the following pde:
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } R^N $$
where $a(x)$ is a smooth bounded vector field ...

**1**

vote

**1**answer

165 views

### Estimates of fractional heat kernel

Is there any estimate available for the derivatives of the fractional heat kernel? Estimates on the kernel itself are at this link.
Also is any estimate available if we consider the problem with ...