# Tagged Questions

**0**

votes

**0**answers

91 views

### Existence of the Dirichlet heat kernel for arbitrary open subsets?

consider first of all an open and bounded subset $\Omega\subset\mathbb{R}^n$, s.t. the boundary $\partial \Omega$ is a manifold of class $C^2$. Then I know that there exists a Dirichlet heat kernel, ...

**3**

votes

**0**answers

71 views

### Is there any weighted sobolev embedding with non-decaying weight

Is there any weighted sobolev embedding like
$$(\int_{R^d}(1+|x|)^s |u|^pdx)^{1/p}\leq C||\nabla^a u||_{L^q}$$?
Here $s>0$, and for some appropriate $p, q$.

**3**

votes

**2**answers

208 views

### Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that
...

**1**

vote

**1**answer

229 views

### A property of groups of operators

Let $X$ be a Banach space. We consider the evolution equation:
$$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$
where $A$ is a bounded operator.
I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, ...

**5**

votes

**0**answers

93 views

### Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...

**0**

votes

**0**answers

61 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**0**

votes

**1**answer

339 views

### Teaching profession:Differential Equations and Mean Value Theorems

Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...

**2**

votes

**0**answers

153 views

### Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions
of the Nonlinear Schrödinger Equation" the following property.
Given the problem
\begin{equation}
\left\{
\begin{array}{rl}
...

**0**

votes

**0**answers

83 views

### Wave operator for focusing NLS

Consider the NLS equation
\begin{equation}
\left\{
\begin{array}{rl}
iu_t + \Delta u+u|u|^{\alpha}=0\\
u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\
\end{array}\right.
\end{equation}
where ...

**5**

votes

**3**answers

158 views

### Method to compute fundamental solutions which are distributions

The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...

**0**

votes

**0**answers

111 views

### Convergence of functions of bounded variation

Let $u_n$ be a sequence of smooth functions with $||u_n||_{W^{1,1}}<C<\infty$. By $BV$ compactness, a subsequence of $u_n$ converges to some $u$ in $L^{n/(n-1)}$. Assume $\int_{\Omega} (|\nabla ...

**5**

votes

**1**answer

393 views

### Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation?

I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f_t + f_{xx}/2 = 0$ for the ...

**2**

votes

**2**answers

158 views

### What is the right initial domain for the Dirichlet-Laplacian on a bounded domain?

Given some bounded domain $\Omega\subset \mathbb{R}^n$ with sufficiently regular boundary (e.g. smooth boundary). Then I saw two slightly different definitions for the Dirichlet-Laplacian.
Some books ...

**4**

votes

**1**answer

213 views

### Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form
$L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$
for ...

**1**

vote

**1**answer

397 views

### Precise versions of “differential operators are unbounded but closed linear operators”

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...

**0**

votes

**1**answer

530 views

### Is a convex function continuous and almost everywhere differentiable? [closed]

is this statement true ?
assume $f:D\rightarrow \mathbf{R} $ is a convex function where $D\subset \mathbf{R}^n$ is a convex set. $f$ is continuous and almost everywhere differentiable and in class ...

**1**

vote

**0**answers

90 views

### growth bound for solution of an ordinary integro-differential equation

I am considering the following ordinary integro-differential equation
$$
A g = \sigma^2(y) + \int_\mathbb{R} \nu(y,dz) z^2
$$
where
$$
A = b(y) \partial + \frac{1}{2} \sigma^2(y) \partial^2
+ ...

**-1**

votes

**1**answer

102 views

### Proving convergence of an integral-differential equation [closed]

I have a second order nonlinear ordinary differential equation which I transformed into an integral-differential equation by multiplying the ODE by $y'$ and integrating.
My question is where can I ...

**1**

vote

**0**answers

263 views

### How is Kolmogorov forward equation derived from the theory of semigroup of operators?

In Lamperti's Stochastic Processes, given
a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$,
a ...

**3**

votes

**2**answers

351 views

### Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$

Hi,
I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$.
Fix $T>0$. ...

**1**

vote

**1**answer

281 views

### Solution of a PDE and its uniqueness

Hallo,
consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE
...

**0**

votes

**0**answers

155 views

### Harmonic Function?

Hi,
Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( ...

**4**

votes

**3**answers

312 views

### ordered exponential of unbounded operators

Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
...

**5**

votes

**0**answers

191 views

### Linear ODEs in a locally convex vector space

Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth ...

**1**

vote

**0**answers

103 views

### The existence of the solution of the perturbed KdV Equation(semi-group operator)

Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...

**5**

votes

**1**answer

382 views

### laplace equation on manifolds with boundary

in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f $ if ...

**3**

votes

**2**answers

383 views

### Do these kernel functions satisfy the semi-group property?

Dear Friends,
Define the kernel functions for $a\ge 1$,
$$
G_a(t,x) := \frac{C_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in R\;,
$$
where the constant $C_a$ is some normalization ...

**2**

votes

**2**answers

388 views

### Will the eigenvalue of the dirac operater tend to negative infinity?

Question: If M is a spin manifold. Condider the dirac operator on a spinor bundle. Can the eigenvalue of this operater tend to negative infinity? If it can, can we choose a riemann matrix such that ...

**3**

votes

**1**answer

575 views

### Long time behavior of the heat equation on R

Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...

**3**

votes

**0**answers

359 views

### PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let ...

**21**

votes

**9**answers

11k views

### Is square of Delta function defined somewhere?

Hello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where?
In the beginning, this question might look strange. But by restricting ...

**2**

votes

**1**answer

303 views

### What is the regularity of the argument of a complex function?

Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...

**6**

votes

**6**answers

778 views

### Application of bounded spectral theory.

I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are ...

**1**

vote

**1**answer

231 views

### variational formulation: boundedness of the bilinear form

The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to ...

**5**

votes

**3**answers

1k views

### Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way

I am interested in a characterization of the creation and annihilation operators that is in some sense invariant under $O(n)$ rotations of $\mathbb{R}^n$:
Background
The Harmonic Oscillator on ...

**0**

votes

**1**answer

282 views

### A differential equation

let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
...

**5**

votes

**4**answers

2k views

### When is Sobolev space a subset of the continuous functions?

If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several ...

**7**

votes

**2**answers

1k views

### Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...

**2**

votes

**3**answers

615 views

### Error analysis of implicit functions

I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial ...