1
vote
1answer
30 views

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 $$f_x\in L^{3/2}(0,\infty).$$
-1
votes
1answer
30 views

Show $W^{1,p}(a,b)$ is compactly embedded in $L^p(a,b)$ for any $1<p\leq \infty$

I always get these sorts of questions wrong. Any help would be appreciated. This is my answer: The problem assumes $(a,b)$ is any bounded interval in $\mathbb R$. Consider a bounded sequence ...
1
vote
0answers
42 views

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 ...
0
votes
0answers
13 views

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. ...
6
votes
1answer
52 views

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 ...
1
vote
1answer
50 views

Sobolev, Holder, Lp spaces continous 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 ...
0
votes
1answer
49 views

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 ...
1
vote
1answer
31 views

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 ...
5
votes
0answers
73 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
2
votes
1answer
46 views

Integral convergence involving Lp and Sobolev spaces

Quick question, any contribution or hint would be appreciated: How does it follow that: $$\lim\limits_{k \rightarrow \infty}\int_{\Omega}a(u_{k})\frac{\partial u_{k}}{\partial x}\frac{\partial ...
3
votes
0answers
43 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
1
vote
1answer
35 views

Implications of Weak convergence in Sobolev Spaces

A quick question regarding weak convergence in Sobolev Spaces. If $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$ for bounded $\Omega$ then can we show that $\nabla u_{k} \rightharpoonup \nabla u$ in ...
4
votes
1answer
59 views

Is $\{f\in H^1;\;\int f=0\}$ dense in $\{f\in L^2;\;\int f=0\}$?

Let $L_*^2=\left\{f\in L^2(a,b);\;\int_a^b f\;dx=0\right\}$ and $H_*^1=\left\{f\in H^1(a,b);\;\int_a^b f\;dx=0\right\}$, where $-\infty<a<b<\infty$. Is $H_*^1$ dense in ...
3
votes
1answer
77 views

Fractional Sobolev embedding into $L^\infty$

Are there any $t\in(0,1)$, $p\in[1,\infty)$ such that $W^{t,p}(\mathbb{R})$ is continuously embedded into $L^\infty(\mathbb{R})$? I have been looking several literatures, but I have not yet found ...
0
votes
1answer
62 views

Bounded subsequence in Sobolev Space

The following is an exercise. Let $I=(0,1)$ and let $(u_n)$ be a bounded sequence in Sobolev space $W^{1,p}$, First question: does "bounded" here means that (for a suitable $M$) $$ \| u_n \|_p ...
3
votes
1answer
27 views

A basic question about $W^{1,1}(\Omega)$

Let $\Omega$ be an open interval $(-1,1)$. Does there exist $u$ in $L^1(\Omega)$ such that $u$ is not in $W^{1,1}(\Omega)$? Let $I=(a,b)$ ba an open interval, possibly unbounded. ...
2
votes
1answer
37 views

Integral convergence and weak convergence

Given that $\Omega \subset \mathbb{R}^{n}$ is a connected bounded Lipshitz domain and $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$. We denote $\Gamma$ as the boundary of the domain. We have the ...
2
votes
1answer
116 views

Show that $\lim_n \|\partial^s (f_n - g_n)\|_p = 0$ (no homework…)

the setting is as follows: Let $\Omega \subset \mathbb{R}^m$ be open and consider some $L^p(\Omega)$ which I will shortly write as just $L^p$ from now on. Furthermore let (for some $k \in ...
2
votes
1answer
77 views

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: ...
4
votes
1answer
55 views

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?
2
votes
2answers
172 views

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 ...
3
votes
1answer
72 views

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 ...
3
votes
1answer
92 views

Gauss–Ostrogradsky formula for Distributions

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 ...
3
votes
1answer
125 views

$W^{1,p}$ compact in $L^\infty$?

Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
3
votes
1answer
93 views

Index of a Fredholm Operator on Paths

I'm a novice to analysis but I need to understand the following example. Any help would be greatly appreciated. This might be of interest to some because it gives a way of quantifying changes in ...
5
votes
1answer
253 views

Is there anyway to bound the $L^\infty$ norm by other $L^p$ norm?

If $f\in L^\infty(\mathbb R^2)$ (in my particular exercise, $f\in H^2(\mathbb R^2)$, the sobolev space), I want to bound $|f|_{L^\infty}= $ esssup $|f|\leq c|f|_{L^p}$ for some p, what kind of number ...
1
vote
0answers
162 views

A continuous embedding.

If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.