# Tagged Questions

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

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### Proof of the Sobolev Space chain rule from Kesavan's Book

I put chain rule on the title because that's what I think they are asking here: This is taken from Kesavan's Functional Analysis book, exercise 2.9 Suppose $\Omega_1$ , $\Omega_2$ are bounded open ...
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### Rellich-Kondrachov

I read an article about the Rellich-Kondrachov embedding theorem in Sobolev spaces: https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem. Nevertheless, when I checked the refererence in ...
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### Sobolev embedding in $C^k(R^d)$.

I wish to prove that for $s > d/2 +k$, where $k \in \mathbb{N}_0$, the Sobolev space $H^{s}(\mathbb{R^{d}})$ is embedded in $C^{k}(\mathbb{R^{d}})$. I found in Folland: http://prntscr.com/bl8kjh ...
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### Sobolev Space dual

I'm interested in the dual space of the Sobolev space $H^1(\Omega)$ for $\Omega$ a bounded smooth domain. Of course, because $H^1(\Omega)$ being a Hilbert space, it's dual is isomorphic to itself, but ...
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### Determine if a function belongs to the sobolev space $W^{1,p}(\mathbb{R})$ and not to $L^q(\mathbb{R})$

I don't understand the first conclusion of the user Tomas in the exercise Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$....
I have a question about a function defined on a Banach space. Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$ and $V:\Omega \to [0,\infty]$ a bounded function on $\Omega$. Let $H^{1}(\... 2answers 29 views ### Can we extend a function$u \in H_0^1(\Omega)$to$\overline{u} \in H_0^1(\widetilde{\Omega})$with$\Omega \subset \widetilde{\Omega}$? Suppose we have$u \in H_0^1(\Omega)$. I want to know if it is always possible to extend it to an open set$\widetilde{\Omega}$such that$\Omega \subset \tilde{\Omega}$by using the extension: $$\... 0answers 26 views ### Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for W^{s,p}(\Omega) when s is an integer Take W^{1,2} = H^1 for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a u\in H^1(\Omega) with the exponent being in integer:$$ ... 1answer 40 views ### If$\{\nabla u_j\}$is Cauchy in$L^p(\mathbb{R}^n)$and$\int_{B(0,1)} u_j dx = 0$, does$\{u_j\}$converge in$L^p_{\text{loc}}(\mathbb{R}^n)$? Let$1 < p < \infty$. Let$\{u_j\}_{j=1}^\infty$be a sequence of functions in$W^{1,p}_{\text{loc}}(\mathbb{R}^n)$such that$\nabla u_j \in L^p(\mathbb{R}^n)$for all$j$,$\int_{B(0,1)} u_j ...
There is a sentence in Evans I cannot justify. The claim made is $u=g$ on $\partial U$ in the trace sense. Why? I understand that $u\in H^1$ implies $u\in W^{1,p}(U)$. But we also need the assumption ...