# 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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### Sobolev space membership of logarithmic function

Determine the largest $s\in(0,1)$ for which the following integral converges $$\int_0^1\int_0^1\frac{\Big|\log|x-\frac{1}{2}|-\log|y-\frac{1}{2}|\Big|}{|x-y|^{1+2s}}^{2}dxdy$$
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### Sobolev Space $W_0^{1,p}(I)$ and the boundary of $I$

Given $I \subset\mathbb{R}$ an open interval, the Sobolev Space $W_0^{1,p}(I)$ is defined as $W_0^{1,p}(I)=\overline{C^1_c(I)}^{W^{1,p}(I)}$ (The closure of $C^1_c(I)$ on the space $W^{1,p}(I)$) . ...
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### Sobolev embedding fails for $p=n$

As everyone knows, the Sobolev embedding fails fails for $n\ge 2$ if we assume $p=n$. The standard example is the function $u(x)=\log \log \bigl(1+\tfrac{1}{x}\bigr)$. This function is obviously ...
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### Find $y \in W_{2}^{1}[-1,1]$ s.t. $\forall x \in W_{2}^{1}[-1,1]$, $f(x)=\langle x, y \rangle$

Consider a Sobolev space $W_{2}^{1}[-1,1]$ with the following inner product: $\langle x, y \rangle = \int_{-1}^{1} [x(t)y(t)+x^{\prime}(t)y^{\prime}(t)]dt$. Let $f(x) = \int_{-1}^{1}e^{2t}x(t)dt$. ...
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### Approximate Sobolev function by smooth function - error estimate?

I wondered if there is, in general, a way to estimate the error (in $L^2$) between a Sobolev function and its mollified version. Let's say $\Omega\subset\mathbb{R}^n$ is bounded with smooth boundary ...
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### Are compactly supported functions in $W^{1,2}(\mathbb R^n)$ also in $W_0^{1,2}(\mathbb R^n)$? See for proof?

The question is stated clearly in the title. On the one hand, it seems obvious (and I give an argument below). On the other hand, after a quick search I haven't been able to find the statement ...
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### vector-valued function space definition except for measure zero

I am wondering what's the correct way to mathematically describe the following problem. Say you have an object that can be defined as an open set $\Omega \in \mathbb{R}^d$, where the dimension $d=2,3$...
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### Poincaré-like inequality

Let $\Omega\subset\mathbb{R}^3$ be an open bounded set. Let $\partial\Omega=\Gamma^1\cup\Gamma^2$, with $\Gamma^1\cap\Gamma^2=\emptyset$. We denote as $\Gamma^1_j$, $j=1,\dots,p_{\Gamma^1}+1$, the ...
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### show that statements are equivalent (Sobolev Spaces)

Given that $u \in L^p(R), 1<p<\infty$ show that the following statements are equivalent: a) $u\in W^{1,p}(R)$ b) $\exists c>0$ such that for $\forall h \in R$ the following inequality ...
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### approximation in $W^{k,p}(\Omega)$ with $C^\infty_c(\Omega)$ functions

I read the following result in real analysis: Let $1 \leq p < \infty$ and $k \geq 0$. Then the space $C^\infty_c({\bf R}^d)$ of test functions is a dense subspace of $W^{k,p}({\bf R}^d).$ A ...
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