Questions tagged [weak-derivatives]

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.

396 questions
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Let $\Omega \subset \mathbb{R}^{N}$ and $W^{1,p}(\Omega)$ be Sobolev Space. Then we let $u\in W^{1,p}(\Omega)$ and define (i) $u^{+} := \max\{0,u\}$ (ii) $u^{-} := \max\{0,-u\}$ Then, we claim that u^... 0answers 17 views Distributional derivative and non-distributional derivative In which points is the following function differentiable (in non-distributional sense)? $$f(t)= t \theta(t) + (1/2t^2-t+1/2)\theta(t-1)+(t-2)\theta(t-2)$$ My solution: \begin{align*} f'(t)&= \... 0answers 24 views Convergence of Mollifier inW_0^{l,p}(\Omega)$I want to prove that for$u \in W_0^{l,p}(\Omega)$we have convergence of mollifiers$u_\rho \rightarrow u$. I appreciate that$u_\rho \rightarrow u$in$L^p(\Omega)$so just need to show convergence ... 1answer 20 views Partial integration on hypersurfaces - Why is there no “boundary integral” In https://homepages.warwick.ac.uk/staff/C.M.Elliott/DziEll13a.pdf on page$299$the notion of weak derivatives on hypersurfaces is introduced via For a$2$-dimensional hypersurface$\Sigma \subset \...
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I have a question concerning an operator based on weak derivatives: Let $\Omega \subset \mathbb R ^3 \times \mathbb R ^3$ be an open and bounded set with smooth boundary and $0 \in \Omega$. Then, the ...
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Is there a version of Taylor's theorem with integral remainder which is valid if the function is only differentiable in a weak sense?

Short question: Is there a version of Taylor's theorem with integral remainder which is valid if the function ($\mathbb R\to\mathbb R$) is only differentiable in a weak sense? Or at least if the ...
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Integrabilty of distibutions

Let $T : D(\Omega) \to \mathbb R$, be a generalised function. What can we say about its integrability? For example, does it belong to $L^p(\mathbb R)$? I am interested in the case of non-regular ...
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Why are test functions in the definition of weak derivatives required to be $\mathcal{C}^{\infty}$?

Recall that a function $u \in L^1_{\text{loc}}(a,b)$ is said to be weakly differentiable with weak derivative $\nu$ if the equation \begin{align} \int_{a}^{b} u(x) \phi'(x) dx = - \int_{a}^{b} ...
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To show a function is in a Sobolev space, can we use weak spherical derivatives?

For example, say $\Omega=B(0,1)$ in $\mathbb{R^2}$, and I have a function represented by $$u(x)=\begin{cases}\ln\ln1/|x|& |x|\leq e^{-2}\\\ln(2) &e^{-2}<|x|\leq 1.\end{cases}$$ There is a ...
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Does the weak derivative satisfy the product rule?

For a locally integrable function $f$, a weak derivative $f'$ satisfies the following relation: $\int{f\phi^{\alpha}}=(-1)^{\alpha}\int{f'\phi}$. Does a weak derivative also satisfy the product rule? ...
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Question about to Weak derivative of $|x|$

As I know that the function $f(x)=|x|$ is not differentiable.but in the weak sense it has weak derivative my question is it again weak derivative exists for this function I.e., suppose $f_1$ is ...
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What is the motivation of defining weak derivative as it is?

I've been reading lately about reproducing kernel Hilbert spaces (RKHS) and Gaussian processes (GP) and during my studies I came across with the concept of weak derivative and Sobolev spaces. I have ...
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Weak differentiability vs differentiability almost everywhere

Is there any relation between weak differentiability and differentiability almost everywhere? Does one imply the other? Does Lipschitz continuity imply both of them? Thank you.
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How is the differential of a Sobolev function on a manifold regarded as an a.e. defined section of $T^*M$?

Let $(M,g)$ be a smooth compact Riemannian manifold, and let $f \in W^{1,p}(M)$ for $p\ge 1$. (I don't assume $p>\dim M$). I have seen in various sources that people refer to the weak derivative ...
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Weak derivative of absolute value of function

Let $\Omega \subseteq \mathbb{R}^n$ be a domain. Suppose $u$ is locally integrable (i.e. $u\in L_{loc}^1(\Omega)$) and has a locally integrable weak derivative $\partial_i u$. Is there a way to find ...
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Elliptic PDE- solution in weak sense

I looking for some literature, where I can find strategy how to solve elliptic PDE in weak sense - define notion of weak solution. Can anyone recommend me something ? Or some links to solved examples. ...