# 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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### Prove that following are true for $\phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$

Fix a function $\phi \in C_c^{\infty}(\mathbb{R}), \phi \ne 0$ and set $u_n(x)=\phi(x+n)$. Let $1 \le p \le \infty$. Then Check that $u_n$ is bounded in $W^{1,p}(\mathbb{R})$ Prove that there ...
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### Spectrum of the Laplace operator with Neumann condition on intervals

Let $-\Delta f=-f''$ be the Laplace operator on $[0,l]$ with domain consisting of functions on $[0,l]$ which have are in $H^2([0,l])$ and satisfy $f'(0)=f'(l)=0$. Then $-\Delta$ is self-adjoint and ...
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### Prove, for an open $\Omega \subset \mathbb{R}^n$ with $x\in \Omega$, that $u\in W^{1,p}(\Omega-\{x\})\implies u\in W^{1,p}(\Omega).$

Let $n\geq 2$, $\Omega\subset \mathbb{R}^n$ open, $x\in \Omega$ and $p\geq1$ I want to prove the above implication. We just need to show that the weak derivative of $u$ on the punctured domain remains ...
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### Construction of Sobolev space

I am reading about the construction of Sobolev spaces from $L^2$. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ...
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### Help showing $u \in W_0^{1,p}(I)$ if and only if $u=0$ on $\partial I$

I am reading the proof the following statement provided in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by haim Brezis: If $u \in W_0^{1,p}(I)$, then $u=0$ on \$\partial ...