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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4
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0answers
79 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
6
votes
1answer
77 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 ...
0
votes
1answer
25 views

Reference needed for: $u \in H^1(0,T;L^2)$ if and only if $\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$

There is a result of the form: a function $u \in H^1(0,T;L^2)$ if and only if $$\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$$ holds for all $h \in [0,T]$. I have only seen one place ...
4
votes
1answer
107 views

Showing Lipschitz continuity of Sobolev function

Is there any problem with the following, please advise: Take $I \subset \mathbb{R}^{n}$ convex, closed and bounded. I want to show that if I have $u_{m} \rightharpoonup^{*} u$ in $W^{1,\infty}(I)$ ...
3
votes
1answer
106 views

Differences between $-\Delta: H_0^1(\Omega)\to H^{-1}(\Omega)$ and $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$

I'll try to explain what I want to know: Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. When we look to $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$, the meaning of $-\Delta$ is ...
1
vote
1answer
59 views

Getting the bound $\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau)| |\nabla u(t+h) - \nabla u(t)|\;dxd\tau dt \leq C$

Let $u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)).$ Is it possible to find the following bound: $$\frac{1}{h}\int_0^{T-h}\int_t^{t+h}\int_\Omega |\nabla u(\tau,x)| |\nabla u(t+h,x) - ...
2
votes
2answers
35 views

Is it possible to estimate $| u |^2_{H^1}$ by $|u|_{H^2}$ for bounded functions?

Let $u\in [L^\infty(\Omega)]^m \cap [H^2(\Omega)]^m$ be a vector valued function with bounded $\Omega \subset \mathbb{R}^n$. Moreover, let $\|u\|_{L^\infty} \leq 1$. Is it possible to bound the square ...
1
vote
1answer
105 views

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 ...
0
votes
0answers
91 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
5
votes
1answer
64 views

Is the gradient operator surjective?

Let $\Omega \subset \mathbb{R}^{n}$ be open and bounded with Lipschitz boundary. Is the gradient operator $\nabla :H^{1} ( \Omega ) \rightarrow L^{2} ( \Omega )$ surjective? Here $H^{1} ( \Omega ) ...
1
vote
1answer
42 views

What is the usual norm of $\mathbb{H_0^1}$?

What is the usual norm of product of sobolev spaces $\mathbb{H_0^1}=H_0^1 \times H_0^1=W^{1,2}\times W^{1,2}$? In my work i need to prove that the norm endowed by the inner product ...
0
votes
1answer
67 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 ...
2
votes
1answer
65 views

Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega:=\mathbb{R}^2\setminus\{0\}$. Consider the $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$ $$ H=-\partial_x^2-\partial_y^2+ ...
0
votes
1answer
26 views

About an estimate of theorem 3 in Chapter 12 of Evans' book

This is the proof of theorem 3 in Chapter 12 of Evans' book as the following picture. I really don't understand why $|F(Du,u_t,u)|\le C(|Du|+|u_t|+|u|)$, because he didn't give us any restriction on ...
1
vote
1answer
31 views

On defining appropriate energy. Any principle?

I am reading Evans' book Partial differential equations. but I am really curious about how he define the appropriate energy? Is there any principle or rule to do this things? Because I notice that ...
3
votes
0answers
57 views

Sobolev Spaces and Convergence

I have a question about one of my homework question. I have been struggling for a while and I really need some help. Assume $N>2$ and $u_k$ is a bounded sequence in $W^{1,2}(\mathbb{R}^N)$ ...
2
votes
0answers
49 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
1
vote
1answer
58 views

Don't understand proof of a PDE argument, author uses $w(t,\cdot) \in L^6(\Omega)$ when $w(t,\cdot) \in H^1(\Omega)$.

I'm reading this paper. I attach the details below(you don't have to read it all to understand my question). My question is, $w$ is only assumed to be $H^1$ in space, why then in the manipulations ...
1
vote
0answers
49 views

implications of convergence in sobolev spaces

If we are given that $O \subset \Omega$ is open and bounded and $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$. We have a sequence $\{u_{m}\}$ satisfying $$ u_{m} ...
0
votes
0answers
26 views

Minimizing the homogenuos Sobolev norm for a given trace

Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of ...
1
vote
1answer
37 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 ...
0
votes
1answer
58 views

Bounding $\int_0^T\int_\Omega v|\nabla u|^2$ given that $v \in L^2(0,T;L^2(\Omega))$ and $u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega))$?

If $v \in L^2(0,T;L^2(\Omega))$ and $$u \in L^2(0,T;H^1(\Omega)) \cap L^\infty(0,T;L^2(\Omega)),\tag{1}$$ is possible to get a bound on the integral $$\int_0^T\int_\Omega v|\nabla u|^2$$ of the form ...
2
votes
1answer
35 views

What would be to show $(1+|\xi|^2)^{s/2}\hat{u}\in \mathcal{S}^{'}(\mathbb R^n)$ is a function?

If $s\in\mathbb R$ define the sobolev space $H^s(\mathbb R^n)$ as $$H^s(\mathbb R^n):=\{u\in \mathcal{S}^{'}(\mathbb R^n): (1+|\xi|^2)^{s/2}\hat{u}\in L^2(\mathbb R^n)\}.$$ I have a doubt concearning ...
4
votes
1answer
61 views

If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?

Let $$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is ...
3
votes
1answer
134 views

The dual space of the Sobolev space $H_0^1$

I am slightly confused about the properties of the dual space of the Sobolev space $H_0^1$ as outlined on page 299 in Evans. In particular, following the notation in the book, item 3 says that ...
1
vote
1answer
38 views

Is $L^2(0,T;H^{-1}(\Omega)) \subset \mathcal{D}^*((0,T)\times \Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be a domain. Consider the space of test functions $\mathcal{D}((0,T)\times \Omega)$ and the space of distributions $\mathcal{D}^*((0,T)\times \Omega).$ Is it true ...
0
votes
1answer
38 views

Is $[L^2(\Omega), H^2(\Omega)]_{\frac 1 2}=H^1(\Omega)$?

Is the interpolated space of order $\frac 1 2$, $$[L^2(\Omega), H^2(\Omega)]_{\frac 1 2}$$ equal to $H^1(\Omega)$? I can't find any good examples of these interpolation ideas. Assume $\Omega$ is ...
5
votes
0answers
88 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
votes
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
votes
1answer
42 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
34 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
votes
1answer
59 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
99 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
votes
0answers
30 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
vote
1answer
38 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
votes
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
66 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
votes
1answer
46 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
votes
0answers
49 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
52 views

Weak star limit in $W^{1,\infty}$

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
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
0answers
23 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
vote
0answers
61 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
0answers
71 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
146 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
51 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
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
33 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
39 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 ...