# 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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### 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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I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), j=1,... 1answer 93 views ### How to prove that$\mathcal{D}(\Omega)$is dense in$\mathcal{D}'(\Omega)$? How to prove that$\mathcal{D}(\Omega)$is dense in$\mathcal{D}'(\Omega)$? Where$\mathcal{D}(\Omega)$is the space of test functions with support compact and$\mathcal{D}'(\Omega)$is the ... 1answer 40 views ### Sufficient conditions that guarantee the integrate by part formula? Suppose, for simplicity, that$f,g$are functions defined on$[0,1]$. Under suitable hypotheses on$f$and$g$, the integrate by parts formula yields $$\int_0^1 f'(t)g(t)dt = f(t)g(t)|_{t=0}^1 - \... 1answer 31 views ### Poincare's inequality with difference quotient For the classical Poincare's inequality, if u \in H^1_0(\Omega), then$$\int_\Omega u^2 \,dx \le C \int_\Omega |\nabla u|^2 \,dx.$$Do we have something similar with the difference quotient? That is,... 1answer 25 views ### Is this an elements of the Sobolev-Space W^{1,2}\left(0,T,H^1(\Omega),H^1(\Omega)'\right)? Definition Let p\in \mathbb{Z}, a<b \in \mathbb{R}, and X, Y be real Hilbert spaces. We define$$ W^{1,p}(a,b,X,Y) := \left\{ \varphi \, \Big| \, \varphi \in L^p(a,b,X),\, \varphi' \in ... 0answers 53 views ### Product of two$W^{1,p}_0$functions I have a fairly simple question. Suppose that$p>n$and$u,v\in W^{1,p}_0$, how can I prove that the product is also in$W^{1,p}_0$? Of course, we have to employ Morrey's inequality. My idea was: ... 0answers 36 views ### Can we show that each element of the Sobolev space$H^k(D)$, with$D\subseteq\mathbb R^d$being a bounded domain, has a continuous representative? Let$d\in\left\{2,3\right\}D\subseteq\mathbb R^d$be a bounded domain$\lambda$be the Lebesgue measure on$DH^k(D)$be the Sobolev space Can we show that each element of$H^k(D)$has a ... 0answers 9 views ### Norms of Slobodekii and classical Sobolev space For r to be a non-negative integer, we have $$\|u\|_{W^{r,p}(\Omega)} = \left( \sum_{|\alpha|\le r} \int_\Omega |\partial^\alpha u(x)|^p dx \right)^{1/p}.$$ For$0 < \mu < 1$, let$s = r + \mu$. ... 0answers 11 views ### Interpolationspace for H^{-1}\cap H^{1} For which$\theta\in[1,\infty)$does hold$(H^{-1}(\Omega),H^1(\Omega))_{1-\frac{1}{\theta},\theta}=L^2(\Omega)$if$\Omega$is a bounded domain with smooth boundary and is three dimensional. I don't ... 0answers 24 views ### Help showing compactness of the support of a function in the Sobolev Space$W^{1,p}$In Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, in the proof of Theorem 8.12, it is needed to show that the support of a function is compact. The function ... 1answer 34 views ### Poincaré-inequality I learned this week about sobolev-spaces and thereby my prof. claimed the truth of an inequality (without proof) that seemed to him very clear and easy to see but to me it was not clear at all. He ... 1answer 102 views ### Show that if$\,u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$, then$\,u \in W_{0}^{1,p}\left(I\right)$. I want to show the following statement ($1 \leq p < \infty$), for an open interval$I$: If$u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$then$u \in W_{0}^{1,p}\left(I\right) $.$W^...
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 ...