For question about weak derivatives, a notion which extends the classical notion of derivative and allows us to consider derivatives of distributions rather than functions.
2
votes
2answers
54 views
Laplace transform of the derivative of the Dirac delta function
$$\int_{0}^{\infty} \delta'(t) e^{-st} \ dt = \delta(t)e^{-st} \Big|_{0}^{\infty} + s \int_{0}^{\infty} \delta(t) e^{-st} \ dt $$
$$= 0 - \lim_{t \to 0} \delta(t)e^{-st} + s e^{-st}\Big|_{t=0} = - ...
1
vote
1answer
39 views
Finding the weak derivative of order $3$ of $f(x)=\operatorname{sgn} \sin(x)$ where $\operatorname{sgn}$ is the sign function
Let $$f(x)=\operatorname{sgn} \sin(x)$$ where $\operatorname{sgn}$ is sign function. I need to find the weak derivative of order 3 for $f(x)$?
1
vote
1answer
40 views
Use difference quotient with not uniform bound to appoximate weak derivative
Suppose U is an open set,not necessarily bounded or has Lipschitz boundary, $f\in L^p(U)$ ,define the difference as usual: $$D^h_i f=\frac{f(x+he_i)-f(x)}{h},\ \ \forall x\in U'\subset\subset U$$
...
1
vote
1answer
32 views
Why is the weak limit of the derivatives the derivative of the weak limit here?
In [1, chapter 8.2.1.b, p.466] the author uses the following argument:
Let $U \subset \mathbb{R}^N$ be an open, bounded domain with smooth boundary. Given a bounded sequence $(u_k)_{k \in ...
5
votes
2answers
93 views
Weakly differentiable but classically nowhere differentiable
Is there any example of a function which is weakly differentiable but none of its versions are classically differentiable (or differentiable only on a set of measure 0) ? Thanks
1
vote
1answer
41 views
Weak Differentiability of Holder functions
Is it true that every Holder function is weakly differentiable? If not please give counterexample. Thanks
4
votes
2answers
106 views
about weak derivative ( Sobolev Spaces )
the following afirmation is true ?
Consider $\Omega $ a bounded and smooth domain . Let $u \in W^{1,p} ( \Omega)$ ( p>1). Supose that $u \geq 0$. let $\alpha >1$ . Then $\nabla u ^{\alpha} = ...
0
votes
0answers
28 views
A weak chainrule [Urbano, Intrinsic Scaling]
Hey I'm reading the book on intrinsic scalign by Urbano an there is a certain issue i have problems with.
Essentially the problem is the following. Let $\Omega\subset \mathbb{R}^n$ be a bounded ...
1
vote
1answer
98 views
Derivative in the sense of distributions
I have a question regarding calculating the derivative in the distribution sense of the following function:
$$
f(x) = \frac{d^2 }{d x^2}|\cos|x||
$$
Maybe someone can point me in the right ...
2
votes
2answers
61 views
Weakly convergence in $W^{1,q}, 1<q<\infty$
Let $(x_n)$ be weakly convergent against $x$ in the Sobolev space $W^{1,q},1<q<\infty$.
Now I have to show, that $(\dot{x_n})$ converges weakly against $\dot{x}$ in $L^q$.
(With the point I ...
3
votes
1answer
113 views
Prove that $f$ does not have a weak derivative
Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by:
$\begin{equation*}
f(x)=\left\{
\begin{array}{rl}0 & \text{if } x\leq 0,\\
1 & \text{if } x\geq 1, \\ 1/2 & \text{if } ...
1
vote
1answer
91 views
Distributional/weak time derivative basic question
Suppose we have $u \in L^2(0,T;H^1(\Omega))$, and $v \in L^2(0,T;H^{-1}(\Omega))$ is the weak time derivative of $u$, so by definition it satisfies
$$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$
...
1
vote
0answers
99 views
When the weak derivative just is the strong (or classical) derivative?
When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
1
vote
1answer
84 views
Weak derivative of $\operatorname{sgn}(x_1)$
Let $x\in \mathbb{R}^{n}, x = (x_1,\ldots,x_n)$, and $f(x) = \operatorname{sgn}(x_{1})$. Is $f$ weakly differentiable on $U = B(0,1)$, i.e. unit ball in $\mathbb{R}^{n}$, and what is the weak ...
1
vote
1answer
222 views
Quick question on definition of derivative in the sense of distribution
Consider a function f such that on $(-\infty,0)$ and $(0,\infty)$, f is differentiable. At 0, there is a point of discontinuity.
e.g. $f(x) = 0$ for $x\leq 0$ and $f(x)=x$ for $x>0$
Then if we ...
1
vote
1answer
168 views
Easy question on derivative in the sense of distribution
I would like help proving this elementary result:
Let $f\in L^{1}_{loc}(a,b)$. Let $x_0 \in (a,b)$ Let $F(x)=\int^{x}_{x_0} f$. Then $F'=f$ in the sense of distributions.
i.e How do I show ...
2
votes
1answer
96 views
Consistency of derivative definitions in Sobolev spaces
Just for the sake of completeness, I begin defining the Sobolev space $H^m(\mathbb{R}^n), \; m \in \mathbb{N}$, as the following set: $H^m(\mathbb{R}^n) = \{u \in L^2 : P^{\alpha} F u \in L^2,\; ...
3
votes
1answer
126 views
Weak derivatives
How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$.
Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
0
votes
0answers
229 views
Weak derivative
Let $u \in C(\Omega)$ be a function with weak derivative $Du \in C(\Omega)^n$. How does one prove that $Du$ coincides with the classical derivative?
Is the mean value theorem for integration ...
