0
votes
1answer
29 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
2
votes
0answers
92 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
1
vote
0answers
102 views

Solution to heat equation, differentiation under integral sign

For $f\in L^1(\mathbb{R}^n)$ a solution to the heat equation $\frac{1}{2}\Delta u=\frac{\partial}{\partial t} u$ is given by $$u(x,t)=(2\pi t)^{-n/2} \int\exp{\left(-\frac{\lVert ...
1
vote
1answer
71 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
1
vote
0answers
24 views

norm of $x \in \mathbb R^d$ is in Sobolev space

For which values of $\alpha, k,p,d$ is $$ \|x\|^\alpha \in \textrm{W}^{k,p} (B(0,1)) \quad ? $$ where $\displaystyle{ \textrm{B}(0,1) = \{x \in \mathbb R^d : \|x\|<1\}}$ This is an ...
1
vote
3answers
99 views

Is there a lower bound for $\int_{B_{r}}f$ when $f$ is a positive function?

Let $f$ be a positive Lebesgue integrable function on a ball $B_{r}\in\mathbb{R}^n$. I'm looking for a positive constant $c$ depending only on $f$ that satisfies $$c\omega_nr^n\leq\int_{B_r}f$$ where ...
3
votes
1answer
109 views

Trace for $L^\infty$ functions?

I'm considering the following problem. Let $s\in L^\infty ((0,T)\times K)$ for some compact $K\subset \mathbb{R^n}$ be given. Consider the Steklov average in time of $s$, i.e. for $h>0$ and ...
1
vote
0answers
124 views

Take the limit of a function inside a Lebesgue integral.

I've been working on the Fundamental solution of Homogenous Heat Equation and I have problem with the following equality. $$\lim_{\epsilon \to 0} \int_{\mathbb{R}} \frac{1}{2\sqrt{\pi}} ...
1
vote
2answers
69 views

Lebesgue integration for $u \in C^{\infty}_c$

Let $u \in C^{\infty}_c(\Bbb{R}^d)$, where $C^{\infty}_c(\Bbb{R}^d)$ is the family of infintly differentiable functions with a compact support. Is $u$ in $L^2(\Bbb{R}^d)$? I think that $u$ is in ...