1
vote
0answers
20 views

A derivative identity, for any multi-index.

I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ ...
0
votes
1answer
23 views

Weak derivatives equals zero

Im just learning sobovel space, I was wondering if the weak derivate holds similar things of the original derivate. Let $U\subset \mathbb{R^n}$ is a open set, and $u\in W^{1,p}$ if $$Du=0 \ \ a.e$$ ...
1
vote
1answer
96 views

Weak Derivative Heaviside function

I have to prove that the Heaviside function $$ H(x):=\begin{cases} 1 &\mbox{if } x \in [0,+\infty) \\ 0 &\mbox{otherwise}\end{cases} $$ doesn't admit weak derivative in ...
1
vote
1answer
29 views

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ make sense?

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ exist, i.e., is $u^+$ weakly differentiable in time? By $u^+$ I mean the positive ...
2
votes
1answer
48 views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T_\epsilon(x) = \begin{cases} ...
0
votes
1answer
31 views

If $u \in L^2(0,T;L^2)$ and $u' \in L^2(0,T;H^{-1})$, is $u \in C^0([0,T];V)$ for some space $V$?

If $u \in L^2(0,T;L^2)$ has weak derivative $u' \in L^2(0,T;H^{-1})$, is $u \in C^0([0,T];V)$ for some Banach space $V$? For what $V$ Lebesgue spaces does this hold? I cannot find any results.
0
votes
0answers
19 views

If $v_i \in L^2(0,T;H^1)$ with $v_i' \in L^2(0,T;H^{-1})$, what can we say about $w(t) = \int_s^t (v_1-v_2)(\tau)\;d\tau$?

If $v_i \in L^2(0,T;H^1)$ with $v_i' \in L^2(0,T;H^{-1})$, what can we say about $w(t) = \int_s^t (v_1-v_2)(\tau)\;d\tau$? Is $w$ also of the same regularity? Heuristically, I thought: let $V_i$ be ...
4
votes
1answer
60 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
0
votes
1answer
29 views

Does weakly differentiable and $L^{\infty}$ imply continuity

Suppose $\Omega \subset \mathbb{R}^d$ is open, connected and bounded. Is $$W^{1,1}(\Omega)\cap L^{\infty}(\Omega) \subset C(\bar{\Omega})?$$ Here $W^{1,1}$ denotes the space of all weakly ...
2
votes
3answers
90 views

Is a function in $L^2$ which second derivative is in $L^2$ in $H^2$?

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary. Assume that $f\in L^2(\Omega)$ and $f^{\prime\prime}\in L^2(\Omega)$. Does one have $f\in H^2(\Omega)$? Useless comments: ...
3
votes
0answers
114 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
0
votes
1answer
37 views

Example of weak derivative on multivariable function

In order to explain about the concept of weak derivatives, I plan to give examples on them. I already manage one example for the single-variable case, but I think it would be better if I can provide ...
2
votes
1answer
62 views

Chain rule for weak derivatives of $f(u)$ where $f'$ is not bounded but $u$ is?

Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$. Suppose $u$ has a weak derivative $u_x$. I want the chain rule $$\partial_x (f(u)) = f'(u)u_x$$ to hold. We know this holds if $f'$ is bounded. But I don't ...
1
vote
1answer
63 views

Notion for weak derivatives of $L^p(0,T,X)$-functions

A definition in Evan's PDE-book from chapter 5.9.2 says (let $X$ be a Banach space): Let $u\in L^1(0,T,X)$. We say $v\in L^1(0,T,X)$ is the weak derivative of $u$ provided $$\int_0^T \phi'(t)u(t)dt = ...
2
votes
2answers
59 views

Density in $H^1_0$

Let $\Omega = \mathbb{R}^2_+=\{(x,y)\in \mathbb{R}^2; y>0\}$ and let $v \in H^1_0(\Omega)$. For $h \neq 0$ we define $D_h v = \dfrac{v(x+h,y)-v(x,y)}{h}$ such as $\forall \varphi \in ...
1
vote
1answer
82 views

Sobolev space-exercice [closed]

Let $\Omega = \mathbb{R}^2_+$. My question is: how we prove that if $v \in H^2(\Omega)$ such as $v(x,0)=0$, then $\dfrac{\partial v}{\partial x} \in H^1_0(\Omega)$ ?
2
votes
0answers
25 views

How to interpret $\langle (f(u))', v \rangle = \langle f'(u)u', v \rangle$?

We know that if $u$ has a weak derivative $u'$, and if $f \in C^1$ with $f'$ bounded then $(f(u))'= f'(u)u'$. But how interpret the duality pairing $$\langle (f(u))', v \rangle = \langle f'(u)u', v ...
1
vote
0answers
20 views

Showing $\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$

Let $u \in L^2(0,T;H^1)$ have weak derivative $u' \in L^2(0,T;H^{-1}).$ Let $c$ be a constant. I want to show that $$\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$$ where $(f)^+ = ...
0
votes
1answer
101 views

Establishing the n-th order weak derivative.

Consider $$AC^{n}:=\{f\in{AC}:f^{(k)}\in{AC}, 1\leq k \leq n-1\}$$ where $AC$ stands for the space of absolutely continuous functions. Now, let $f,g\in{L_{loc}^{1}(a,b)}$ and ...
5
votes
2answers
250 views

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
1
vote
2answers
79 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
3
votes
1answer
230 views

Weak Derivatives and Lp spaces

I just want to know how to prove two properties of weak derivatives and $L^{p}$ spaces if they are true, the first one just involves weak derivatives: If we have a locally summable function $u: U ...
1
vote
0answers
56 views

Self-adjointness of differentiation operator

Let's say $\mathcal{H}=L^2([0,1])$ and $p$ is the operator $-i\frac{d}{dx}$ defined on $\mathcal{D}(p)=\{f\in L^2([0,1])\ |\ f'\in L^2, f(1)=e^{i\theta}f(0)\}$. I have to prove that $p$ is ...
2
votes
1answer
147 views

Has a probability density function a weak derivative

Assume I have a probability density function $\rho$ on $R$. (e.g $\rho \geq 0$ $\int \rho dx =1 $ $\rho \in L^1(R)$ ...). So $\rho$ is the density wrt the lebesgue measure. Now I try to understand if ...
0
votes
0answers
55 views

Chainrule: “piecewise smooth” and Sobolev functions

since we had a lengthy discussion on my last question (Generalized chainrule for Sobolev functions with a cut-off I didn't find an answer to it yet, I'm going to post a related more specific question ...
2
votes
0answers
111 views

Generalized chainrule for Sobolev functions with a cut-off

let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $f\in C^1(\bar\Omega \times (0,g(x)),[0,1])$ and $f(x,\cdot)$ increasing and $g(x)\in\mathbb{R}$ continuous (maybe better, Lipschitz?). I want ...
1
vote
0answers
58 views

Does existence of weak spatial derivative imply existence of classical time derivative in this situation?

Let $f(x_1,...,x_n,t)$ be a function, where $(x_1,...,x_n) \in \mathbb{R}^n$ and $t \in [0,T].$ Denote by $f_{x_i}$ the weak (partial) derivative of $f$ wrt. $x_i.$ Is it possible for ...
1
vote
0answers
124 views

If a sequence of smooth functions converge in the Sobolev norm, what can one say about the limiting function?

Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of $k$ times weakly differentiable functions on $\mathbb{R}^n$, having finite Sobolev $p$ norm. Suppose $\{f_i\}$ is a Cauchy sequence in ...
1
vote
1answer
151 views

Steklov Average in time and spatial gradient can be interchanged?

I'm trying to understand a proof on Steklov average and weak derivatives. Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $T>0$, $h\neq 0$ and $u\in L^2(0,T;H^1(\Omega))$ and extend $u$ by ...
7
votes
1answer
349 views

Are a weak derivatives and distributional derivatives are different?

For simplicity, given a real function $f\in L^1_{loc}(\Omega)$, we define both weak or distributional derivatives by $\int f'\phi = - \int f \phi'$ for all test functions $\phi$. Now, take $\Omega = ...
3
votes
2answers
191 views

Differences between $C_c^\infty[0,T]$ and $C_c^\infty(0,T)$

I believe it is true that: If $f \in C_c^\infty(0,T)$, then $f(T)=f(0)=0$. $C_c^\infty(0,T) \subset C_c^\infty[0,T]$ $C^\infty(0,T) \subset C_c^\infty[0,T]$ If $f \in C_c^\infty[0,T]$, it doesn't ...
2
votes
2answers
220 views

Weak / Classical derivative

I know the definitions of both weak and classical derivative. But I am trying to see the classical derivative as a weak derivative. When we have $\int f' \varphi = -\int f\varphi'$ for all $\varphi\in ...
1
vote
1answer
263 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 ...
1
vote
1answer
149 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
320 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 ...