0
votes
0answers
20 views

Convergence in $L^2$ of difference quotients to derivative of function in $H^1$

Is it true that if $u\in H^1({\mathbb R})$, then $(u(x+h)-u(x))/h$ converges to $u'(x)$ in $L^2({\mathbb R})$, as $h\to 0$? It's hard for me to get a handle on this, since $u'$ doesn't have to be ...
0
votes
1answer
53 views

What is the Frechet derivative of $(u^+)^q$?

I know that if we define $E[u]=\int_\Omega u^+dx$, where $\Omega$ is compact in $R^n$ and $u\in H_0^1(\Omega)$, $u^+:=\max\{u,0\}$, then $E[u]$ is not Frechet differentiable. However, if now I define ...
0
votes
0answers
8 views

Bessel Potential spaces

Let $\Omega_1,\Omega_2 \subset \mathbb{R}$ be bounded. The mapping $F: \Omega_1 \rightarrow \Omega_2$ shall be bijective, continuously differentiable and such that $||DF(x)||$ and $||(DF(x))^{-1}||$ ...
1
vote
1answer
41 views

Strong differentiability in Sobolev spaces

My question is: how can one prove that if $\phi\in H^{n}(a,b)$ for all positive integer $n$, then $\phi\in C^{\infty}(a,b)$? $H^{n}(a,b)$ denotes de Sobolev space of the real interval $(a,b)$. For my ...
0
votes
1answer
48 views

Mean value theorem in sobolov space under integral

Sorry this seems like a basic question, but I'm having trouble figuring out the answer. Let g(x) be the step function over [-1,1] and f(x) a function with $f\in H^1[-1,1]$, that is it has a square ...
1
vote
1answer
81 views

Critical exponent in O.D.E and Gâteaux derivative

I have this example and I don't understand its resolution: Let $\Omega \subset \mathbb{R}^n , n\geq3$ be a bounded open set (with smooth boundary), let $f\colon\mathbb{R}\rightarrow \mathbb{R}$ be ...
1
vote
1answer
64 views

Identify the distrionbutional derivative with classical derivative?

I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma. In proving the theorem, he defines the function $F$, and calculates its ...
8
votes
2answers
480 views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?