**2**

votes

**1**answer

343 views

### The PDE $u_t=u_{xx}-u_{yy}$: The simplest linear second-order PDE that isn't elliptic, parabolic, or hyperbolic.

I know that there have been several questions on here and stackexchange about linear PDE's which don't fall into the standard classification, but I had a more focused question which I haven't seen ...

**3**

votes

**0**answers

148 views

### Exponential decay for the gradient of a solution

Dear all,
I would like to prove the exponential decay of the derivatives of a solution to the following equation in $\mathbb{R}^N$:
$$
\sqrt{-\Delta+m^2} u +u= f(u),
$$
where I can assume that $m \neq ...

**15**

votes

**1**answer

500 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
$$
...

**8**

votes

**2**answers

737 views

### PDE with the Jacobian Determinant

Hello,
Could you please help me in answering the following question?
Initially I thought that the following problem can be solved through Monge-Ampere equation, but with Monge-Ampere, I have not ...

**3**

votes

**1**answer

234 views

### Hormander's bracket condition for the adjoint of an operator

Let $X_0, X_1, \dots, X_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator
$$
L = \sum_{i=1}^k X_i^2 + X_0~.
$$
Here, I assume that Hörmander's bracket condition is ...

**1**

vote

**1**answer

251 views

### Please recommend some classical books or articles on the compressible Euler equations !

Please recommend some classical books or articles on the local well-posedness result of compressible Euler equations ! The main aim is that I want to learn some basic methods and techniques about the ...

**3**

votes

**1**answer

186 views

### C^\infty versus semiclassical wavefront sets

Zworski states that if $u$ is a compactly supported distribution, independent of the semiclassical parameter $h$, then the relationship between the $C^\infty$ and semiclassical wavefront sets of $u$ ...

**5**

votes

**0**answers

95 views

### Lagrangean uniqueness versus Eulerian uniqueness

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE:
$$
\dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N.
$$
...

**0**

votes

**0**answers

115 views

### damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation.
My question is...why is the ...

**2**

votes

**1**answer

135 views

### Distributional limits concerning the regularity of Maxwells equations

This question is related to my previous question about the regularity of the Maxwell equations.
Assume we are working on a space where there are only electric point charges, $(q_i)$, and a blob of ...

**74**

votes

**16**answers

5k 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 ...

**5**

votes

**3**answers

391 views

### Divergence form Elliptic PDE Removable Singularity/Regularity Question

Idea
Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...

**5**

votes

**1**answer

350 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

661 views

### fixed point arguments in PDE

I was curious whether anyone knows of some examples in PDE where
a standard fixed point argument fails to show the existence of a solution but one can apply
one of the more advanced fixed point ...

**1**

vote

**1**answer

240 views

### Does this PDE has a general solution? [closed]

$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$
K and u are constants.
If no,how to get a asymptotic solution?ie,linearize

**0**

votes

**1**answer

409 views

### Showing a coercivity condition for this bilinear form

Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to ...

**3**

votes

**3**answers

277 views

### Is there a good metric under which a sequence of compact sets can converge to an infinite dimensional set?

I have a sequence of finite-dimensional, compact sets in $L^2(\Omega)$, where $\Omega\subset \mathbf{R}^2$ is closed and bounded. The dimension grows monotonically with the sequence, and there is no ...

**1**

vote

**2**answers

201 views

### Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...

**9**

votes

**1**answer

621 views

### Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf ...

**4**

votes

**1**answer

525 views

### Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello!
Let's say I have
$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$
with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).
Now separate the Amplitude and Phase of the solution:
...

**8**

votes

**1**answer

491 views

### The Speed of Gravitational Waves in General Relativity

Is it possible to mathematically prove that the speed of gravitational waves in general relativity equals the speed of light, without linearizing the Einstein Field Equations? The approach via the ...

**1**

vote

**1**answer

65 views

### Finding singular “solutions” to the Dirichlet problem for Schrödinger operators that do not admit smooth solutions

Suppose that $\Omega\subset \mathbb{R}^n$ is a smooth open region and that $V:\Omega\to \mathbb{R}^+$ be a positive smooth function.
Then we have a family of operators
$$L_\epsilon =-\Delta -\epsilon ...

**2**

votes

**1**answer

277 views

### Change in neumann boundary conditions through coordinate transformation of elliptic PDE, weak formulation

The standard weak formulation of the Neumann problem for the Poisson equation
is to find $u \in H^1 ( \Omega)$ such that for every $v \in H^1 ( \Omega)$:
$$ \int_{\Omega} \nabla u \nabla v d x = ...

**1**

vote

**0**answers

149 views

### two polynomial equations

Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system
$$f_{11}+2tf_{12}+t^2f_{22}=0$$
$$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$
...

**3**

votes

**1**answer

152 views

### Subharmonic envelope

I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ...

**6**

votes

**1**answer

314 views

### Green functions on Riemann surfaces

Let $(M,g)$ be a compact Rieamnnian surface without boundary and $\Delta_g$ be the Lapalce operator. We note $\lambda_i$ and $\phi_i$ the eigenvalues and eigenunctions of $\Delta_g$. Let also $G_g$ ...

**1**

vote

**1**answer

155 views

### Well-posedness of Euler-Poisson system for semiconductors

Is there anyone can recommend me some important literature references about the well-posedness of both Cauchy problem and initial boundary value problem of Euler-Poisson system for semiconductors? ...

**0**

votes

**0**answers

167 views

### hitting probability for integrated Ornstein-Uhlenbeck process

Consider an Ornstein-Uhlenbeck position process:
$dV_t=dB_t-\lambda V_tdt$
$dX_t=V_tdt$
where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ .
Let $r>0$ and $S_r$ be the ...

**2**

votes

**2**answers

409 views

### Easy question on Sobolev spaces

I understand that this question would be trivial for experts, sorry for that, I just need to clarify things.
So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W_p$, $W_q$ are ...

**2**

votes

**1**answer

299 views

### Weak divergence implies weak differentiability of components?

Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$.
We say that $\sigma$ has weak divergence if there exists ...

**1**

vote

**1**answer

162 views

### A Cauchy problem for an iterated Euler-Poisson-Darboux eqaution

Good morning,
I'm interested in solving a Cauchy problem for the iterated singular EPD.
Well, Weinstein (On a class of PDEs of even order, 1955) showed how the decomposition formula leads to the ...

**0**

votes

**0**answers

81 views

### About the boundedness of the derivative of a function which is in a special function space.

If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, $f (t,x) : bounded\; on \; [0,T] \times \Bbb R^n $ then how can I conclude that
$$ \left \| \frac{\partial f}{\partial t} \right ...

**6**

votes

**0**answers

273 views

### Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by
$$\frac{\partial}{\partial t} ...

**4**

votes

**2**answers

244 views

### Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform

Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose
$$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$
Is $a(x,0)=0$, $\forall x\in[0,1]$?
...

**2**

votes

**1**answer

149 views

### Reference request: Stability / instability theory for periodic orbits of partial differential equations

I am looking for references regarding the stability / instability of a periodic solution to a partial differential equation / evolution equation in infinite dimensions. Suppose we have a periodic ...

**8**

votes

**2**answers

1k views

### what's the idea behind Carleman estimate

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

**1**

vote

**2**answers

317 views

### Variational problems whose lagrangian density depends on derivatives higher than 1.

The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...

**0**

votes

**1**answer

542 views

### The conormal derivative of a function

Hi!
I was wondering about the definition of the conormal derivative of a function
$u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered
as functionals on ...

**1**

vote

**1**answer

348 views

### Elliptic Differential Equations with rough boundary data

Question stated roughly:
Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one ...

**4**

votes

**1**answer

133 views

### Interpretation of a parameter in forming a pseudodifferential operator

In Zworski's Semiclassical Analysis, he defines the following method of quantization: for a symbol $a = a(x,\xi) \in \mathscr{S}(\mathbb{R}^{2n})$ and $u \in \mathscr{S}(\mathbb{R}^n)$,
$$ ...

**2**

votes

**1**answer

200 views

### Solvability of a nonlinear elliptic equation

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation
\begin{equation}
-\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0.
...

**2**

votes

**3**answers

875 views

### Jacobi method on first order partial differential equations

Hi,
I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject.
Thanks in advance

**3**

votes

**1**answer

187 views

### Reference Request: Schauder theory for fourth-order parabolic equations

I am looking for a treatment of fourth order parabolic equations in Holder spaces. More precisely fourth order analogues of Theorems 5.1, 5.2, and 10.1 in Chapter IV of Linear and quasilinear ...

**5**

votes

**3**answers

798 views

### Reference request: parabolic PDE

I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics.
I think I have a firm grip ...

**6**

votes

**4**answers

616 views

### Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be smooth compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by
\begin{equation} ...

**2**

votes

**1**answer

298 views

### Nash inequality on a compact domain?

I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the ...

**4**

votes

**1**answer

590 views

### Solving PDE with Cauchy - Kowalewski Theorem

Hallo,
I have the following PDE that I am trying to solve via the Cauchy-Kowalewski Theorem. But I have no idea how to do it or if its possible. Maybe one of you has an idea. Here is the problem: Let ...

**2**

votes

**1**answer

206 views

### A question on Schwartz distributions

I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...

**0**

votes

**1**answer

144 views

### On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
...

**4**

votes

**3**answers

881 views

### Quasi-linear System of First Order P.D.E.s of “Mixed” type

In my research work, I am dealing with a quasi-linear system of first order p.d.e.'s with two independent variables (say $x_1$ and $x_2$) and four dependent variables (say $u_1(x_1,x_2)$, ...