# Tagged Questions

Sobolev spaces are function spaces generalising the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, functions in a Sobolev space have also differentiability conditions: that is, we require all partial derivatives of the function up to a ...

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### What is the Sobolev Lemma?

In the paper I am reading the authors state that $|\nabla u|_\infty$ can be replaced by $|u|_3$ using the Sobolev Lemma. I am trying to find this lemma but its turned out to be very difficult. The ...
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### Showing that smoothing operators are compact

Suppose I have a bounded, linear map $T: H^1(X) \to H^1(X)$ such that $T(H^1(X)) \subset C^\infty(X)$. Is $T$ a compact operator? I'm guessing this depends on whether or not $X$ is (pre)compact, and ...
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### Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
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### Mean value theorem in sobolov space under integral

Sorry this seems like a basic question, but I'm having trouble figuring out the answer. Let g(x) be the step function over [-1,1] and f(x) a function with $f\in H^1[-1,1]$, that is it has a square ...
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### How can we glue Sobolev functions?

Let $u:A\cup B\to R$ be a function (where $A$ and $B$ are disjoint sets and $A\cup B$ is connected) such that $u$ restrict to $A$ and to $B$ are in $W^{1, p}$. Which result guarantees me that $u$ is ...
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### Reference of Sobolev space

currently I'm using Krylov's book, while consulting Evans (too many details are left out, for my level). Also, Adams 1975 version has been widely cited. So besides these ones, which book in your ...
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### What is “approximation in a Sobolev Space”? For example,

I want to know the meaning of the statement as below. $$\text{There is a sequence} \;\{f_k\} \subset W^{3,2}\; \text{approximating}\;\; f \;\;\text{in}\;\; W^{2,2}.$$ Here $W^{n,m}$ means a ...
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### When Dirac function is in $H^{-m}(R^n)$?

If Dirac function $\delta\in H^{-m}(R^n)$,please give the range of $m$?
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### There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$?

There exist a function $u(x)\in C([0,1]),~ u(0)=u(1)=0$ such that $u \notin H_{0}^{1}((0,1))$? Justify your answer! Thanks.
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### When does $v \in H_0^1$ and $|v|_{L^2}=0$ imply that $\|v\|_{H_0^1}=0$?

Let us assume that the boundary of the domain in the definition of the Sobolev spaces $L^2$ and $H_0^1$ is sufficiently smooth. Let $|\cdot |$ denote the norm in $L^2$. Then for a function $v$ in ...
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### Show reflexivity of Sobolevspace $W^{1,4}(0,1)$

I would like to show elementary - using the canonical embedding - that the Sobolevspace $W^{1,4}(0,1)$ is reflexive. Therefore I set $X=W^{1,4}(0,1)$ and now I have to show that the canonical ...
### if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then$f\in H^k (\Omega')$
prove that if the region $\Omega'\subset\Omega$ and $f\in H^k (\Omega)$ then $f\in H^k (\Omega')$
### An inequality : $\| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$
Let $u=u(x)$ be a real-valued function defined on $\mathbb R$. How does this inequality hold? $$\| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$$ ...