1
vote
2answers
35 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
2
votes
1answer
40 views

Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
2
votes
0answers
29 views

Alternative derivation of Poincaré inequality

I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is ...
2
votes
1answer
38 views

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 ...
3
votes
3answers
90 views

Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
1
vote
1answer
61 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
1
vote
1answer
31 views

Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
0
votes
1answer
135 views

Is this proposition about $L^2$ functions correct?

Is this proposition correct? Will you please give a contour example if it is wrong? If $J \in L^2(\mathbb{R}) \cap C^1 (\mathbb{R})$, $f \in C^{\infty}(\mathbb{R})$, $|f'|\leq K$, where $K > 0$ is ...
0
votes
2answers
23 views

Inner Product Representation of Functional on $H^1_0$

Let $T\colon H^1_0(\mathbb{R})\to \mathbb{R}, \;T(f)=\int f'\phi \;dx$, where $\phi\in L^2$ is fixed. By Hölder, $|T(f)|\leq\|\phi\|_2\|f'\|_2\leq C \|f\|_{H^1_0}$, i.e. $T$ is continuous. Therefore, ...
2
votes
2answers
40 views

$f\in W^{1,p}((0,1))$ is absolute continuous

This is an exercise in Evans's PDE book $f\in W^{1,p}((0,1))$ is absolute continuous where $1\leq p < \infty $ Try : By definition of sobolev space, $f$ has weak derivative $f'$ So $$\ast\ ...
0
votes
2answers
37 views

estimating $L^p$ norm of $\frac{x}{|x|^3} \ast (-\text{div}_x \ldots)$

I've been having problems with following an argument in a paper I'm reading, I hope someone can help me understand the point. Suppose $f(t,\cdot,\cdot)$ is a $C_0^1 (\mathbb{R}^6)$ function for all ...
3
votes
1answer
34 views

Smooth function composed with sobolev function vanishes at 0

Let $\Omega$ be a bounded domain with sufficiently smooth boundary. Let $u \in W^{1, 2}_{0}(\Omega)$ and $F \in C^{\infty}(\mathbb{R} \rightarrow \mathbb{R})$ such that $F(u(x)) = 0$ for almost every ...
2
votes
2answers
54 views

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ ...
0
votes
2answers
26 views

Quadratic Minimization

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If ...
5
votes
1answer
169 views

Stokes theorem and Sobolev spaces.

I am interested under which regularity condition is Stokes' theorem is still valid. For concreteness I am interested in the following problem Let's consider a domain $\Omega$ in $\mathbb{R}^{3}$ ...
0
votes
0answers
28 views

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 ...
1
vote
2answers
83 views

Discontinuous Sobolev Function

I'm trying to show that there's an $f \in H^1(\mathbb{R}^2)$ which is not ae equal to a continuous function. Per a couple of suggestions, I've decided to look at the function $f(x) = ...
1
vote
1answer
34 views

Sobolev inequality in negative index

For $s>n/2$, is it true that $$ \int |fg| dx\leq ||f||_{H^s}||g||_{H^{-s}}?$$ This inequality is used on pg 398 of the Majda Bertozzi book on Vorticity and Incompressible flow but I can't make ...
2
votes
0answers
37 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
2
votes
1answer
88 views

Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
0
votes
1answer
147 views

weak convergence of product of weakly and strongly convergent $L^{2}$ sequences in $L^{2}$

there is one question bothering me for quite a while now. Let $a_{n},b_{n}\in L^{2}:a_{n}\stackrel{L^{2}}{\rightharpoonup} a\in L^{2} $ weakly $ b_{n}\stackrel{L^{2}}{\rightarrow} b \in L^{2}$ ...
2
votes
1answer
110 views

Proof or counterexample: $L^p$-boundedness gives a.e. convergent subsequence?

Let $\Omega\subset\mathbb{R}^{d}$ open and let $f_{n}\in L^{2}\left(\Omega\right)$ be bounded. Then there is obviously a weakly convergent subsequence. Is there also a subsequence converging almost ...
0
votes
1answer
63 views

A version of Rellich-Kondrachov's theorem

Let $D$ be a bounded open subset with smooth boundary in $\mathbb{R}^n$ , $k$ be positive integer, and $p \in [1,\infty)$ such that $kp < n$. Let $q\in[1,\dfrac{np}{n-kp}) $ and put $T(u) = u$ ...
0
votes
1answer
31 views

Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$

Let $D$ be an open subset of $\mathbb{R}^n$ , $p$ and $q$ be in $(1,\infty)$ such that $p^ {-1} +q^ {-1} = 1$. Let $u \in W^{1,p} (D)$ and $v \in W^{1,q} (D)$ . Then $uv$ belongs to $ W^{1,1} (D)$ ...
4
votes
3answers
132 views

Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $\Delta u+\cos u=0$.

Let $D$ be the open bounded smooth subset in $\mathbb{R}^{n}$. Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $$\Delta u+\cos u=0.$$ Help me some hints to start. ...
4
votes
2answers
105 views

Prove that $\int_{D}\nabla u\cdot\nabla vdx=\int_{D}uv\,dx=0$

Let $D$ be the open bounded subset in $\mathbb{R}^{n}$ with smooth boundary, $\alpha$ and $\beta$ be different non-null real numbers, and $u$ and $v$ be in $W_0^{1,2}(D)\setminus\left\{ 0\right\} $ ...
4
votes
2answers
89 views

Let $\Omega$ be a bounded open subset of $\mathbb{R^3}$, and $f$ be in $L^2(\Omega)$ Does there exist a weak solution in $W^{1,2}_0(\Omega)$

Let $\Omega$ be a bounded open subset of $\mathbb{R^3}$, and $f$ be in $L^2(\Omega)$. Does there exist a weak solution in $W^{1,2}_0(\Omega)$ to the following equation: \begin{cases} \Delta ...
1
vote
1answer
81 views

Inequality for function in certain Sobolev space

I have to prove the following inequality for a function $u$ in $H^1(\mathbb{R}^3)$: $$\int_{B_r}\vert u\vert^q\leq C\bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^a\bigg(\int_{B_r}\vert ...
3
votes
1answer
136 views

If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ ...
3
votes
1answer
54 views

Is $f \in W^{1,1}[a,b]$ equivalent to $f$ absolutely continuous on $[a,b]$?

$f$ is a function defined on $[a,b]$. Then $f \in W^{1,1}$ is equivalent to $f$ is absolutely continuous?
1
vote
0answers
35 views

Aproximating a function on SO(3)

By $SO(3)$ I mean rotation matrices. Let $\cal{L}=\{f:[0,l]\to \rm{SO}(3)\} \cap L^1([0,l];\mathbb{R}^{3\times3})$. How to approximate funkctions from $\cal{L}$ with functions from ...
1
vote
1answer
69 views

weak differentiability of log log function

I want to understand why the following function has a weak derivative in two or three dimensions: $w(x) = \ln |\ln|x|| , x \in B_{1/2}(0)$. Can I say that if I have a strong derivative (except for ...
1
vote
1answer
52 views

Convergence of integral means of the gradient of a Sobolev function

Let $B_R(x_0)\subset\mathbb{R}^n$ with $R<1$ for $n\geq3$ and suppose $u\in H^1\big(B_R(x_0);\mathbb{R}^N\big)\cap L^{\infty}\big(B_R(x_0)\big)$. Define, \begin{equation} \phi(R)\equiv ...
1
vote
1answer
93 views

property of local Sobolev space

The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if ...
0
votes
1answer
42 views

limit of function in Sobolev space

Let function $f(x)$ and $f(x)g(x)$ belong to $\mathcal{W}^{s+1}$ with $s\ge 1$, where $\mathcal{W}^{s+1}$ be the Sobolev space of regularity $s+1$ in $\mathbb{L}^2$-norm. We also have $g\in ...
3
votes
1answer
80 views

Similar to Poincare inequality on Sobolev spaces

The following looks quite similar to Poincare's inequality: Let $\displaystyle{ 1 \leq p < \infty}$ and $\displaystyle{ U \subset \mathbb R^n}$ open and such that $\displaystyle{ U \subset ...
0
votes
0answers
30 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
2
votes
1answer
57 views

Range of the Dirac operator on the real line closed?

This is quite a short question: consider the Dirac operator $-i \tfrac{d}{dx}\colon H^1(R) \to L^2(R)$, where $H^1(R)$ denotes the Sobolev space of square-integrable functions with square-integrable ...
0
votes
0answers
38 views

A real analysis question [duplicate]

Let $u,v:\Omega\subset\mathbb{R}^{N} \to \mathbb{R}$ ($\Omega$ is bounded), $v\geq u>0$ (and $u$, $v$ are nice). Is it true that $$ \frac{\displaystyle\int\limits_\Omega ...
3
votes
1answer
136 views

A real analysis question

I would like to ask if the following statement is true or not: Let $u,v:\Omega\subset% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{N}\rightarrow% %TCIMACRO{\U{211d} }% ...
1
vote
0answers
54 views

Continuous and dense embeddings and the density of sets in Hilbert space.

Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose ...
2
votes
1answer
77 views

Is the inclusion map in the Sobolev embedding theorem a surjective map?

Let $W^{k,p}(\mathbb{R}^n)$ be the Sobolev space of all real valued functions on $\mathbb{R}^n$ whose first $k$ weak derivatives are in $L^p(\mathbb{R}^n)$. Assume that $$ \frac{1}{q} = \frac{1}{p} ...
1
vote
0answers
22 views

construction of a sequence of smooth function [duplicate]

Consider $r>0$ , $1<p< \infty $. Consider $K $ a compact set in $R^n$ with $K \subset B(x_0 , r)$. Define $$ B = \{ u \in C^{\infty}_{0}(B(x_0 , r)) ; u=1 \ on \ K , 0 \leq u \leq 1\}$$ ...
1
vote
1answer
46 views

equality involving smooth functions (capacity theory)

This implication is true ? I believe that is .. Consider $E$ a compact subset of $R^n$. $1< q < p$ . Supoose that for all $\Omega \subset R^n$ ($\Omega$ open )occurs ...
2
votes
1answer
74 views

construction of a smooth function using mollifiers

let $r>0$ and $B(x_0, r) \subset R^n$ . My problem is construct a function $u \in C^{\infty}_{0}(B(x_0, 2r))$ using mollification satisfying $$u = 1 \text{ on } \overline{B(x_0, r)} $$ and $$ ...
2
votes
0answers
57 views

weak derivative and the value of a integral

Let $0 < r < R$ and $p>1$ and consider the function $$u(x) = \displaystyle\frac{\displaystyle\int_{|x|}^{R} t^{-1 }dt}{\displaystyle\int_{r}^{R} t^{-1 }dt},$$ if $r < |x|< R$ , and ...
6
votes
2answers
523 views

Product rule of weak derivatives

I am working on proving the following proposition: If $u,v\in {W^1(\Omega)}$ and $uv,uDv+vDu\in L^1_{\operatorname{loc}}(\Omega)$, then we have the product formula $$D(uv)=uDv+vDu.$$ The definition I ...
1
vote
1answer
129 views

Is tensor product of Sobolev spaces dense?

My question is: is $W_2^k(\mathbb{R})\otimes W_2^k(\mathbb{R})$ dense in $W_2^k(\mathbb{R}^2)$, and more generally is this true in $\mathbb{R}^d$? I found this post: Tensor products of functions ...
1
vote
1answer
42 views

Is there a function $u$ with $g(u)$ regular whose “truncation” $g(T(u))$ is not regular?

I hope the title is not to misleading. Assume you have a continuous function $g:R\to R$ and a Lebesgue measurable function $u:\Omega\to R$ for some bounded domain $\Omega$ such that $g(u)\in ...
8
votes
1answer
134 views

Calculus on the Sobolev space valued function of one real variable $t$?

Now I am interested in the calculus on Banach space valued function, especially the function with value in a certain Sobolev space. I want to prove that $$\bigcap_{k=0}^m ...