# Tagged Questions

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### A derivative identity, for any multi-index.

I'm trying to prove by induction on the multi-index $\alpha$, that, $$\sum\limits_{j=\frac{|\alpha|}{2}}^{|\alpha|}\sum\limits_{|\beta|=2j-|\alpha|}c_{\beta}x^{\alpha}[m_z(x)]^{j+1}=D^{\alpha}m_z(x)$$ ...
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### 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 $$Du=0 \ \ a.e$$ ...
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### 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 ...
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### 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 ...
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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 $\forall \varphi \in ... 1answer 82 views ### Sobolev space-exercice [closed] Let$\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)$? 0answers 25 views ### How to interpret$\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 $$\langle (f(u))', v \rangle = \langle f'(u)u', v ... 0answers 20 views ### Showing \langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle Let u \in L^2(0,T;H^1) have weak derivative u' \in L^2(0,T;H^{-1}). Let c be a constant. I want to show that$$\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$$where (f)^+ = ... 1answer 101 views ### Establishing the n-th order weak derivative. Consider$$AC^{n}:=\{f\in{AC}:f^{(k)}\in{AC}, 1\leq k \leq n-1\}$$where AC stands for the space of absolutely continuous functions. Now, let f,g\in{L_{loc}^{1}(a,b)} and ... 2answers 250 views ### Questions about weak derivatives There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ... 2answers 79 views ### Distributional derivatives on hypersurface? In a paper I was reading, the define a set Q=(0,T)\times \Omega, where \Omega \subset \mathbb{R}^n is a bounded domain, and then they write$$\langle \frac{d}{dt}u - \Delta u, \varphi ... 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$ ...
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 ...