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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5
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0answers
109 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 ...
0
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1answer
18 views

Is $C^\infty([0,T]\times \Gamma) \subset C^\infty([0,T];H^1(\Gamma))$? If so, is it dense?

Let $\Gamma$ be a $(n-1)$-dimensional compact hypersurface (with whatever smoothness is required). Is it true that $$C^\infty([0,T]\times \Gamma) \subset C^\infty([0,T];H^1(\Gamma))$$ holds? I'm not ...
2
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1answer
47 views

Trace defined in terms of integral averages

It is known that if $u \in W^{1,1}(U)$ where $U\subset \mathbb{R}^n$ is bounded and $\partial U$ Lipschitz then $\mathcal{H}^{n-1}$ a.e we have $$\lim_{r\to 0} \frac{1}{|B(x,r)\cap U|}\int_{B(x,r)\cap ...
1
vote
1answer
36 views

Transferring a result in PDE from open domain to a manifold

Can anyone recommend me something that in detail, talks about transferring a result in Sobolev space (as opposed to Holder spaces or something like that) that holds for open domains in $\mathbb{R}^n$ ...
2
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1answer
67 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 ...
2
votes
1answer
112 views

Why are so many inequalities and estimate in PDEs?

I want to study PDEs, but when I read some PDEs books, I notice that there are so many inequalities here and always do estimates. I really want to know why there are so many inequalities here and ...
0
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0answers
33 views

inequality for linear functions in sobolev space

Ist the following Statement true for $f$ and $g$ linear? $\vert fg \vert_{H^2} \leq C \Vert f \Vert_{H^1} \Vert g \Vert_{H^1}$, where $\vert \cdot \vert_{H^2}$ denotes the seminorm. My Idea: It is ...
1
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1answer
40 views

Is $\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$ equivalent to $\lVert u \rVert_{H^2(M)}$?

On a bounded Riemannian manifold without boundary, is it true that the norms $$\lVert u \rVert_{L^2(M)} + \lVert \Delta u \rVert_{L^2(M)}$$ is equivalent to the full $H^2$ norm $\lVert u ...
0
votes
1answer
65 views

How to show $\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla v|^2 \varphi \varphi_t - 2\int_Q (\nabla v \cdot \nabla \varphi) (\varphi v_t)$

Let $Q=\bigcup_{t \in (0,T)}\Omega \times \{t\}$. I have seen this identity for all $\varphi \in C_c^\infty(Q)$ such that $0 \leq \varphi \leq 1$: $$\int_Q \varphi^2 (\Delta v)v_t = \int_Q |\nabla ...
0
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1answer
31 views

For a PDE $u' + Au = f$, if $f$ and $u'$ are smooth does it mean $Au$ is also smooth?

Suppose I have a solution $u \in L^2(0,T;H^1(\Omega))$ with $u' \in L^2(0,T;H^{-1}(\Omega))$ of the PDE $$u' + Au = f$$ where $A:L^2(0,T;H^1(\Omega)) \to L^2(0,T;H^{-1}(\Omega))$ is an elliptic ...
2
votes
1answer
77 views

Going from a weak formulation to a pointwise a.e statement; don't understand text (PDEs, sobolev spaces)

I just read this: For $u \in H^1(Q)$ where $Q=\cup_{t \in (0,T)}\Omega \times \{t\}$, we have that $$\int_{\Omega}u_tv +\nabla u \cdot \nabla v = \int_{\Omega}fv\quad\text{for all $v \in ...
3
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1answer
49 views

Sobolev spaces documentation

Can someone indicate some documentation on this subject? (thoroughly explained with a presentation of its applications - mostly interested in the FEM. However just a good presentation of the Sobolev ...
0
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0answers
62 views

Sobolev spaces in polar coordinates

I need some properties about Sobolev spaces in polar coordinates. To be precise, let $U = \{(x,y)\in\mathbb R^2 : x^2 + y^2 < R\}$ be an open disc and let $H_0^1(U)$ be the usual Sobolev space ...
3
votes
1answer
64 views

Predual of $W^{1,\infty}$

I understand the meaning of $u_n$ converges to $u$ weak star (it means that $u_n\in E^*$ and $(u_n,x)_{E^*,E} \to (u,x)_{E^*,E}$ for all $x\in E$) but I've some trouble for identifying a space $E$ ...
2
votes
1answer
91 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
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0answers
25 views

Regularising a function that is constant on an interval (related to Heaviside)

Define the function $f:\mathbb R \to \mathbb R$ by $$f(x) = \begin{cases} x &\text{for $x < 0$}\\ 0 &\text{for $x \in [0,1]$}\\ x-1 &\text{for $x > 1$}& \end{cases} $$ Note that ...
1
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0answers
64 views

some sobolev norm estimation

I would like to show this inequality. I need help to show this inequality Let $F(\Phi)=\left|\Phi\right|^{\alpha}\Phi$ with even integer $\alpha>0$. Let $k$ be a positive integer satisfying ...
3
votes
1answer
87 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 $\mathcal{S}(\mathbb{R}^d)$ ...
0
votes
1answer
290 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
53 views

Sobolevspace on non open set

in the definition of Wikipedia and several books the Sobolevspace $W^{k,p}(\Omega)$ is defined on a open subset $\Omega\subset \mathbb{R}^d$. Why does $\Omega$ have to be open? Why is [0,1] not ...
2
votes
1answer
139 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
34 views

Is $(H_0^1,\|\cdot\|_{L^2})$ a closed subspace of $L^2$?

Let $-\infty<a<b<\infty$ and $f\in L^2(a,b)$. Suppose $(f_n)$ is a sequence in $H_0^1(a,b)$ such that $\|f_n-f\|_{L^2}\overset{n\to\infty}{\longrightarrow}0$. Can we conclude that $f\in ...
0
votes
2answers
41 views

Intuituve affirmation of functions of $H^1_0(\Omega)$

Consider $\Omega$ a open and bounded set. Let $u \in H^{1}_0(\Omega)$ a continuous function. is true that $lim_{x \rightarrow y} u(x) = 0$ for $y \in \partial \Omega$ ? I dont know how to prove ...
0
votes
0answers
25 views

Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...
1
vote
1answer
45 views

Situation of Nirenberg-Sobolev embedding

Suppose $f\in L^2(\mathbb{R}^2)$ with compact support and $\frac{\partial ^{(k)}} {\partial x^{K}}f, \frac{\partial ^{(k)}} {\partial y^{K}}f\in L^2(\mathbb{R}) \; \forall k\in\mathbb{N}$. Can we ...
2
votes
1answer
65 views

simple question about the level set of a function of $H^{1}$

Consider a smooth ,convex and bounded domain $K \subset \{ x_1 = 0 \} \subset R^n$ . consider a smooth open set $\Omega \subset R^{n}_+ = \{ x = (x_1,..,x_n)\in R^n ; x_1 > 0\} $ with $K \subset ...
0
votes
1answer
54 views

If $u$ solves Dirichlet problem $-\Delta u =f$, what is known about $fu$?

Let $u \in H^1_0(\Omega)$ be the weak solution of $$-\Delta u = f$$ $$u|_{\partial\Omega} = 0$$ Is there anything known about the sign of $fu$ a.e?
3
votes
0answers
56 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
2
votes
1answer
97 views

Proving regularity of a function [closed]

Let $\Omega \subset \mathbb{R}$, bounded and regular. Prove that if $u \in H^1(\Omega)$, then $|u| \in H^1(\Omega)$? $H^1(\Omega)=\{u \in L^2(\Omega) \mbox{ s.t } \partial_x u \in L^2(\Omega)\}$
0
votes
1answer
55 views

Basic Query On Sobolev Space

Okay, I am very new to the premise of Sobolev Spaces and there is one exercise here that's really grinding my gears. The premise is this- to what Sobolev spaces for each real number $\alpha$ does ...
2
votes
1answer
74 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 ...
0
votes
1answer
60 views

Common orthogonal basis for $L^2$ and $H^1$

How can we obtain a common orthogonal basis for the space $L^2(U)$ and $H^1(U)$ for some bounded open subset of $\mathbb{R}^n$? That this can be done is mentioned in Evans's Partial Differential ...
2
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1answer
98 views

Showing a linear map has a unique extension from a Sobolev space to an Lp space

This is related to another question I asked: Directional derivative in a Sobolev-like inequality Suppose that $u \in C_{0}^{\infty}(\Omega)$. Show that the linear map $u \to u(x,0) \in ...
1
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1answer
65 views

Don't understand a passage in Showalter (PDEs, sobolev spaces, dual spaces)

Please see this: How does the last equality follow? Why does the duality pairing become an integral over $G$ when test functions are restricted to $L^p$???
0
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2answers
61 views

Getting a bound on solution of PDE in $L^\infty(0,T;L^2(\Omega))$?

Let $$\varphi(s) = \begin{cases} s &: s < 0\\ 0 &: s \in [0,1]\\ s-1 &: s > 1 \end{cases}$$ Note that $\varphi$ is Lipschitz. Consider where $f \in L^2(0,T;L^2(\Omega)$, $$u_t - ...
1
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0answers
29 views

Compactness in Sobolev spaces

I am looking for characterizations of compactness in the Sobolev space $H^{-1}$. In particular, I am looking for a characterization involving the Fourier transform. Can anyone suggest some results ...
2
votes
1answer
108 views

$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required

Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where ...
0
votes
1answer
50 views

solving a equation by variational method

I am studying the proof of the following theorem of this article http://projecteuclid.org/download/pdf_1/euclid.cmp/1103922134 The theorem states: Theorem: For $0 < \sigma < \frac{2}{N-2}$, ...
0
votes
2answers
40 views

weak derivatives of exp(-|x|) and Hilbert Spaces

To which Hilbert Space (W^m,2) of R does the function exp(-|x|) belongs? I know its weak derivative is (-exp(-x) for x>0, exp(x) for x<0 and c0 (arbitrary) for x = 0). This weak derivative is in ...
3
votes
1answer
146 views

Equivalent norms on a cartesian product of Hilbert spaces.

Notation: $H_0^1=H_0^1(a,b)$, where $-\infty<a<b<\infty$. Let $\|\cdot\|_V$ be a norm on $V:=H_0^1\times H^1_0\times H^1_0$ given by ...
0
votes
1answer
76 views

About a compact imbedding of Sobolev spaces

I am studying the Compactness lemma ( on page 570) of the article http://projecteuclid.org/euclid.cmp/1103922134. The lemma says (Compactness lemma ): for $0 < \sigma < \frac{2}{N-2}$, $(N ...
2
votes
2answers
88 views

Directional derivative in a Sobolev-like inequality

I am trying to do the following problem: Let $\Omega \subset \subset \overline{\mathbb{R}_{+}^{n+1}} = \{ (x, x_{n+1}); \, x_{n+1}\geq 0\}$ (i.e., $\Omega$ is bounded inside the closed upper ...
0
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1answer
58 views

Basic question about weak/strong convergence

Let $0<\sigma< \frac{2}{N-2}$ with $N \geq 3$. I know that $H^{1}_{\operatorname{rad}}(R^n)$ (radial functions of $H^{1}(R^n)$ ) is compactly embedded in $L^{2 \sigma +2}(R^N)$. Let $(\psi_v )$ ...
2
votes
1answer
121 views

Example of linear parabolic PDE that blows up

Does anyone have an example of a linear parabolic PDE that blows up in finite time in a Sobolev space setting? How does one show blow up for that particular example? The one in Evans unfortunately is ...
0
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0answers
36 views

Simple question about the Gagliardo Niremberg interpolation inequality

Consider the Gagliardo Niremberg interpolation inequality : (Gagliardo Niremberg interpolation inequality)Let $q,r$ be any numbers satisfying $1 \leq q, r \leq \infty $ and let $j,m$ be any ...
1
vote
1answer
42 views

What does $W^{1,p}(I)\subset C\big(\overline{I}\big)$ mean?

Let $I\subset\mathbb{R}$ be a bounded open interval. Brézis book states that the injection $W^{1,p}(I)\subset C(\overline{I})$ is compact for all $1<p\leq\infty$. Elements of $W^{1,p}(I)$ are ...
4
votes
1answer
63 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 ...
0
votes
1answer
66 views

Is $H^2\cap H^1_0$ dense in $H_0^1$?

Let $-\infty<a<b<+\infty$ and $I=(a,b)$. Consider $H^1_0(I)$ equipped with the norm $\|\cdot\|_{H^1}$ given by $$\|f\|_{H^1}=\|f\|_{L^2}+\|f'\|_{L^2}.$$ Is $H^2(I)\cap H_0^1(I)$ dense in ...
0
votes
1answer
41 views

$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Let $-\infty<a<b<\infty$ and $f,g\in H^1(a,b)$. So, $f,f',g,g'\in L^2(a,b)$. Suppose $$\int_a^b|f+g'|^2\mathrm dx=\int_a^b|f'-g|^2\mathrm dx=0.$$ Is it possible to conclude that $f=g=0$ ...
5
votes
2answers
206 views

PDE uniqueness by energy method contradicts non-uniqueness???

Consider $$u_t - \Delta u + u = 0$$ $$\frac{\partial u}{\partial \nu} = 0$$ $$u(T) = u(0)$$ on a domain $\Omega$ (the BC is obviously on $\partial\Omega$. If $u$ solves this PDE, clearly as does ...