# 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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### assigning boundary values to a weakly differentiable function

There is a sentence in Evans I cannot justify. The claim made is $u=g$ on $\partial U$ in the trace sense. Why? I understand that $u\in H^1$ implies $u\in W^{1,p}(U)$. But we also need the assumption ...
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### Schwartz functions dense?

I want to show that the Schwartz functions are dense in $$\left\{f \in L^2; \int |x|^2 \left|f(x)\right|^2 dx + \int |\xi|^2 \left|\hat{f}(\xi)\right|^2 d \xi < \infty\right\}$$ where the norm ...
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### Is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to $\int_{\mathbb{R}^n}| \nabla \varphi|^2dx$ contained in $L^2(\mathbb{R}^n)$?

Equip $C_0^\infty(\mathbb{R}^n)$ with the norm $$\|\varphi\|^2_1 := \int_{\mathbb{R}^n}| \nabla \varphi|^2dx.$$ Indeed, $\| \cdot \|_1$ is a norm on $C_0^\infty(\mathbb{R}^n)$ because any constant ...
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### If $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$ with $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ then $u_k \to u$ in $W^{1,p}(\bar{\Omega})$

I have already proven that for every $u \in W^{1,p}(\Omega)$ with $1 \leq p < \infty$ and every open set of class $C^1$ there exists a sequence $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ such that ...
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### Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
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### About test functions for supersolutions

Let $B_{1}$ the unit open ball in $\mathbb{R}^{n}$ and $u \in H^{1}(B_{1})$. For $k,m >0$, let $\bar{u} = u^{+} + k$ and $\bar{u}_{m} = \bar{u}$ if $u < m$ and $\bar{u}_{m} = k+m$ if $u \geq m$. ...
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### How fast can Sobolev functions grow?

It is a simple fact that $L^p$-functions cannot grow arbitrarily fast. More precisely, one has for every $\ell>0$ $$|\{f\geq\ell\}|\leq \frac{\|f\|_{L^p}^p}{\ell^p}$$ for every $f\in L^p$. My ...
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### Bound first order derivative by $L^2$ norm of elliptic elliptic operator

Consider an symmetric 2nd order differential operator on a bounded domain with smooth boundary $$A=-\sum_{i,j=1}^n \partial_j (a^{ij}(x)\partial_i)$$ be uniformly elliptic if there exists $C_0>0$ ...
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### How to show a piecewise quadratic interpolant is $H^1$

I am preparing for a final exam and came across this question: Suppose that $\Omega\subset\mathbb{R}^2$ is an open bounded domain with triangulation $\mathscr{T}$. Suppose that $v_h$ is a ...
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### Passing from Classical Formulation to Weak Forulation on PDEs. Integration by parts in $n$ dimensions?

I am reading about the Variational Method for solving PDEs and ODEs. The process to pass from a classical solution to a weak solution is pretty clear when we are dealing with ODEs, by using ...
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### Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
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### Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$

Show that there exists a unique $v_0 \in H^1(0,1)$ such that $u(0)=\int_0^1(u'v_0'+uv_0), \forall u \in H^1(0,1)$. Further Show that $v_0$ is the solution of some differential equation with ...
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### Representation of the delta distribution as an element of the dual of $H^1$

I'm working with some Sobolev spaces and I just wanted to consider the elements of $H^{-1}$ as elements on $H^1$ (Riez Theorem). Since the delta function $\delta(f) = f(0)$ is an element of the dual ...
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### Show that the following are equivalent for $u \in C^{1}(0,1)$

Let $u \in C^{1}(0,1)$. Prove that the following are equivalent: (a) $u \in W^{1,1}(0,1)$ (b) $u' \in L^{1}(0,1)$ (where $u'$ means the derivative in the usual sense) (c) $u \in BV(0,1)$ My try: ...
Consider a connected open bounded subset $D\subset \mathbb{R}^d$ with smooth boundary. It is easier to state for the disc so I will do so. I don't think it would change the argument (except for in our ...