# Questions tagged [sobolev-spaces]

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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### Do we have $\varphi=0$ a.e.?

Suppose we have a sequence of smooth functions with compact support: $\{\varphi_n\}\subseteq C_c^{\infty}(\Omega)$, here $\Omega\subseteq \mathbb{R}^n，n\geqslant 2$ is open. Suppose additionally we ...
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### $C^\infty_{0}(\bar\Omega)$ dense in $W^{k,p}(\Omega)$: the closure is necessary?

It is a fundamental result of Sobolve space that Let $\Omega \subset \mathbb{R}^d$ or $\mathbb{R}^d_+$, then $C^\infty_{0}(\bar\Omega)$ is dense in $W^{k,p}(\Omega)$. However, in some ...
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### “Poincaré” inequality for $H^1$

I have to show the following: Let $U\subset \mathbb{R}^n$ be nice (i.e. bounded, open and boundary of class $C^1$). Further there's a function $$f:H^1(U) \to \mathbb{R}^n$$ which is continuous and ...
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### dense subspaces in Hilbert spaces

So far I know, the Schwartz space $S(\mathbb{R}^n)$ is dense in $L^2(\mathbb{R}^n)$ with respect to the usual $\lVert \cdot \rVert_{L^2}$-norm, i.e., for every function $f \in L^2(\mathbb{R}^n)$ there ...
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### how can I show that $H_0^k(\Omega)=\{u\in H^k(M):\text{supp }u\subset\overline{\Omega}\}?$

Let $\bar{\Omega}$ be a smooth, compact manifold with boundary; we denote the interior by $\Omega$. We can suppose $\bar{\Omega}$ is contained in a compact, smooth manifold $M$, with $\partial\Omega$ ...
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### Under which circumstances is the Laplacian compact?

I want to know when the Laplacian is a compact Operator. Do you know some good literature about this topic? For instance, is the Laplacian compact on the Sobolev space $H^2(\Omega)$? Or maybe on the ...
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### Sobolev space and equivalence of norms

I'm considering the space $W\{n,p\}[0,1]$ of functions with $n-1$ continuous derivatives $f^{(n-1)}$ is absolutely continuous and $f^{(n)}$ is in $L^p[0,1]$. The usual norm is the sum of the $p$-norms ...
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### Eigenfunction associated to the smallest eigenvalue of an elliptic operator

Let $(T_n)$ be a sequence of elliptic operators defined in $H^2(\Omega)\cap H_0^1(\Omega)$ to $L^2(\Omega)$, with $\Omega$ being a bounded domain with smooth boundary. All of them have a smallest ...
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### Equivalent Norms on Sobolev Spaces

Let $u$ be in the Sobolev space $H^2$. Then the standard definition of the norm is $$||u||_{H^2}^2 = \sum_{|\alpha| \leq 2} ||D^{\alpha} u||_{L^2}^2.$$ However, I have also seen it defined this ...
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### Show properties of elements of $\mathcal{H}^2$

I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found ...
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### Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way : Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...