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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3
votes
1answer
38 views

Identification of $L^2$ limits with distributional convergence

I just read the thread on "too much effort" and I would like to be more specific. Is the following reasoning correct: Let $g,g_\delta\in H^1(D)$, $D$ some domain in $\mathbb{R}^n$ with the following ...
1
vote
0answers
49 views

Extending by zero a Sobolev function

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in W_0^{1,2}(\Omega)$. Define $B_R=B(x_0,R)$ for $x_0\in\partial\Omega$ and consider $\tilde{u}=u\chi_{\Omega\cap B_{2R}}$. Do we need some ...
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 ...
-1
votes
1answer
34 views

Show that this functional is coercive - variational methods

For $u \in H^{1}_0({\Omega})$ ($\Omega$ is a domain open and bounded in $R^n$). Let $0 < \lambda < \lambda_1$ ($\lambda_1$ is the first eigen value of the laplacean) and a fixed $f \in ...
2
votes
1answer
70 views

Weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc...
2
votes
0answers
28 views

Dual space of Sobolev functions with homogenous neumann BC

Let $\Omega$ be a bounded Lipschitz-domain with outward normal vector $\nu$ and let us take a look at the Sobolev-spaces $H^1:=H^1(\Omega)$, $H^1_0:=H^1_0(\Omega)$ and ...
1
vote
2answers
40 views

Show that a map is a continuous bilinear form on $H^1(0,1)$ space

Let $u,v \in H^1(0,1) = \{f : (0,1) \longrightarrow \mathbb{R}, f,f' \in L^2(0,1) \}$, show that $$a(u,v) = \int_0^1 (u'v' + uv)\; dx$$ is a continuous bilinear form.
0
votes
1answer
48 views

If $f \in H^1$ and $f=g$ a.e. or in $L^2$, is $g \in H^1$?

If $f \in H^1(\Omega)$ and either $$f=g \quad\text{a.e.}$$ or $$f=g \quad\text{in $L^2(\Omega)$},$$ is $g \in H^1(\Omega)$? I think since we identify functions that are equal almost everywhere that ...
1
vote
1answer
35 views

Does $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ give something useful?

If $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ is there any way to extract a strongly convergent subsequence of $u_n$ in some useful space for PDEs? Or ...
2
votes
2answers
48 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
1
vote
0answers
20 views

Showing $\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$

Let $u \in L^2(0,T;H^1)$ have weak derivative $u' \in L^2(0,T;H^{-1}).$ Let $c$ be a constant. I want to show that $$\langle ((u-c)^+)', (u-c)^+ \rangle = \langle u', (u-c)^+ \rangle$$ where $(f)^+ = ...
1
vote
1answer
36 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
0
votes
1answer
25 views

Bounding an integral (Sobolev spaces and duality pairing), explanation needed

I don't understand how the inequality is derived here: I understand the equality but not how he gets the last line. I get a triple integral and I don't know to deal with that.
4
votes
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
83 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
109 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
109 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
36 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
109 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
94 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
43 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
68 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
59 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
38 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
59 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 ...
4
votes
1answer
146 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
40 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
91 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
100 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 ...