# Tagged Questions

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### Is $C^\infty_0$ dense in $C^\infty$ w.r.t. $\|\cdot\|_{L^p}$ and $\|\cdot\|_{W^{1,p}}$?

Is the space $C^\infty_0(\Omega)$ of smooth functions with compact support, dense in the set of smooth functions $C^\infty(\Omega)$ with respect to the norms $\|\cdot\|_{L^p}$ and the Sobolev-Norm ...
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### What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
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### Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}.$$ Let $\{\Omega_n\}$ be a sequence of open ...
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### Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
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### negative Sobolev space contains $L^1$ for a compact domain

I'd like to use something like Aubin-Lions lemma for the following spaces: $$C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball ...
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### If $u_n$ is bounded and pointwise convergent, then $u_n$ convges in $W_{2,p}$.

I'm reading this paper about solving semilinear elliptic pde's through iterated approximations. The line i'm trying to understand is "Then, since $u_k = Tu_{k-1}$ and since $\{u_k\}$ is a bounded, ...
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### Question about injection on an unbounded space

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$ how to prove that ...
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### Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
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### Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
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### Is 'f' belong sobolev?

I was trying to show that the function $$f(x) = \dfrac{x^{1/2}}{1+x^2} \in W^{1,3/2} (0,\infty)$$ that is, have to show that $$f\in L^{3/2}(0,\infty)$$ and $$f_x\in L^{3/2}(0,\infty).$$
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### Don't understand a PDE argument involving $L^p$ norms and inequalities

I'm reading this paper. I do not understand how the author proves Corollary 5.12 for the case $p=2$. He addresses everything except $p=2$ but claims it holds for all $p$. Can someone help me to see ...
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### Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
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### Proof of equivalence of Sobolev Space and Lipschitz functions

The attachment is a proof from Evans book "Measure Theory and Fine Properties of Functions" pg 132 Theorem 5. The statement of the theorem is: Let $f:U \rightarrow \mathbb{R}$. Then $f$ is locally ...
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### Sobolev, Holder, Lp spaces continuous and compact embeddings proof

I would like to know if the following proof is fine. I haven't filled in all the detail but please let me know what you think about the basic outline.(I am aware that there are posts which have dealt ...
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### Compactness of Sobolev Space in L infinty

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can't directly use the compactness of Rellich Kondrachov ...
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### Is there any Banach space $X$ that $L^2(\Omega)$ is compactly embedded into?

Let $\Omega \subset \mathbb{R}^n$. Is there a good (*) Banach space $X$ that $L^2(\Omega)$ is compactly embedded into: $$L^2(\Omega) \subset\!\subset X$$? If not compactly embedding, I at least would ...
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### Explanation on a “different” proof that $C_c(\Omega)$ is dense in $L^p(\Omega)$.

Theorem: Let $\Omega\subset \mathbb{R}^n$ be an open set and $1\leq p < \infty$. The space $C_c(\Omega)$ is dense in $L^p(\Omega)$. Haim Brezis has a French book called "Analyse fonctionnelle: ...
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### Is it true that $2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$?

For any $C^1$ function defined in $(0,1)$, is it true that $$2^p\|f\|^p_{L_p}+2^p\|f'\|^p_{L_p} >2\|f\|^p_{L_\infty}$$ If it is true, how to prove it?
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### Proof of Sobolev Inequality Theroem

I have a short question about the proof of Theorem 2 below. I have included Theorem 1's statement since it is used in the proof of Theorem 2. Definition: If $1 \leq p < n$, the Sobolev Conjugate ...
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### Find a function in $H^{\frac{1}{2}}$ that is not in $L^{\infty}$.

Les $\mathcal{S}$ be the Schwartz class and $\mathcal{S}'$ be its dual (also known as the set of tempered distributions). For a function $u$ let $\hat{u}$ denote de Frourier transform of $u$. Given a ...
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...