Questions tagged [sobolev-spaces]
For questions about or related to Sobolev spaces, which are function spaces equipped with a norm that controls both a function and its weak derivatives in some Lebesgue space.
4,778
questions
2
votes
1
answer
23
views
Equivalent definitions of Poincare inequality
I can't seem to find this anywhere but I can find that there are (at least) two definitions of the Poincare inequality. One is
$$
\int_\Omega |f|^2 dx \leq c\int_\Omega |\nabla f(x)|^2 dx
$$
for $f \...
- 87
1
vote
1
answer
39
views
Brezis book, Functional analysis, Sobolev spaces and PDE, problem 8.30
Let $k \in \mathbb{R}$, $k \neq 1$, consider the space $$V= \{ v \in H^1(0,1): v(0) = k v(1)\}$$ and the bilinear form $$B(u,v) = \int_0^1 \left(u'v' + uv\right) ~dx - \left(\int_0^1 u\right) \left(\...
- 41
0
votes
0
answers
29
views
Apply Arzela-Ascoli theorem to unifomly bounded sequence in $H^1$
Let $\{u_{i}\}$ be sequence of smooth function defined on $\Bbb{R}^n$ such that $\|u_i\|_{L^2(\Bbb{R}^n)}$ is uniformly bounded in $i$ and $\|\nabla u_{i} \|_{L^{^2}(\Bbb{R}^n)}$ is also uniformly ...
- 2,928
1
vote
0
answers
15
views
How to the estimate as application of Strichartz Estimates?
RecallA pair $(q,r)$ is admissible if $q\geq 2, r\geq 2$ and
$\frac{2}{q} = N \left( \frac{1}{2} - \frac{1}{r} \right),(q,r,N)\neq(\infty,2,2).$
Strichartz estimates Let $\phi \in L^2(\mathbb R^N),$...
- 91
1
vote
0
answers
25
views
Elliptic regularity with right-hand side in $H^{-1/2}$
If $\mathbf{f} \in H^{1/2}(\Omega)$ for a bounded domain $\Omega$ with smooth boundary, does the elliptic regularity for the Laplacian guarantee that the solution to the elliptic problem in divergence ...
- 88
1
vote
0
answers
23
views
Weak convergence in $H^1(\mathbb{R}^N)$ implies weak convergente in $H^1(B)$
Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite ...
1
vote
1
answer
34
views
Proof of the embedding of time dependent Sobolev spaces
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
- 567
0
votes
1
answer
23
views
Hölder continuity on Sobolev space $H^1(a,b)$
Let $H^{1}(a, b)=W^{1,2}(a,b)$ be the Sobolev space, that is, the functions $f\in L^2$ such that $f' \in L^2$, where $f'$ denotes the weak derivative of $f$. This is equivalent to
$$f(x)=c+\int_a^x f'...
2
votes
0
answers
37
views
Proving $||f||^2_{L^2(\mathbb R^2)}\le 10||f||_{L^1(\mathbb R^2)}||\nabla f||_{L^2(\mathbb R^2)}$ with $f\in C^1,L^1\cap L^2$ and $\nabla f\in L^2$.
Prove that there exists a universal constant $K<10$, for all $C^1$ function $f : \mathbb R^2 \rightarrow\mathbb R$, if $f \in L^1 (\mathbb R^2)\cap L^2(\mathbb R^2)$ and $|\nabla f| \in L^2(\mathbb ...
- 804
4
votes
1
answer
43
views
Prove the uniqueness of $u\in H_0^1(\Omega)$ with $\Delta u=\vert u\vert^{q-1}u+f$ in $\Omega$ with $\Omega$ as a bounded domain with smooth boundary.
Problem: Let $\Omega\subset\mathbb R^2$ be a bounded domain with smooth boundary. Prove that, for all $p>1$ and $1\le q<\infty$, for all $f \in L^p(\Omega)$, there exists a unique $u\in H_0^1(\...
- 804
1
vote
0
answers
74
views
+50
Showing $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ is equivalent to $\|u\|_{H^2}$ norm for $H^2$ space
This question has been asked here showing $\lVert \Delta u \rVert_{L^2(U)} + \lVert u \rVert_{L^2(U)}$ is equivalent to $\|u\|_{H^2(U)}$ norm for $H^2(U)$ space assuming the domain $U$ is sufficient ...
- 2,928
0
votes
1
answer
42
views
Elliptic regularity for Poisson equation
Let $U \subset \Bbb{R}^n$ be an open bounded smooth domain, let $u \in H^1_0(U)$ satisfy the Poisson equation:
$$-\Delta u = f \tag{*}$$
with $f \in L^{2}(U)$, by the classical elliptic regularity ...
- 2,928
0
votes
0
answers
32
views
Some questions in the compactness of Sobolev space and Holder space
Let $B_1$ be the unit ball in $\mathbb{R}^2$. Define $A=\{u \in W^{1,2}(B_1) : \|u\|_{ W^{1,2} } \leq 1\}$, $B= \{u \in L^{2}(B_1) : \|u\|_{ L^2 } \leq 1\}$, $C=\{u \in C^{0,\alpha}(B_1) : \|u\|_{ ...
- 523
1
vote
1
answer
25
views
Sobolev inequality for cubes
Consider the fallowing result from Evans, 2010, page 279:
Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ is $C^1$. Assume $1 \leq p < n$, and $u \in W^{1,p}(U)$. Then ...
3
votes
2
answers
92
views
Is it possible to find the maximum of $r\ln\left(\ln\left(1+\frac{1}{r}\right)\right)$ for $r\in(0,1)$?
I was working on Chapter 5, Problem 14 of Evan's Partial Differential Equations 2nd ed and to prove integrability of $\ln \left (\ln \left (1+\dfrac{1}{|x|}\right )\right )$ on the unit $n$-ball, ...
- 51
0
votes
1
answer
16
views
What is the relationship between weak derivatives and the decay of the coefficients in the eigenfunction expansion?
Suppose we have a function $f: [0,1] \to \mathbb{R}$ that has $2\beta$, $\beta \in \mathbb{N}$, weak derivatives. Defining $e_k(x) := \sqrt{2}\sin(k \pi x)$ the orthonormal sine-basis of $L^2([0,1])$, ...
2
votes
1
answer
45
views
Relationship between Sobolev-Slobodeckij spaces and Besov spaces
I am trying to understand these two different ways of defining fractional Sobolev spaces. In particular, I want to determine embeddings or equality between the Besov spaces $B^{s}_{p,p}$ and the ...
- 5,402
-3
votes
0
answers
41
views
Weak derivative of power function
Let $f\in {W}^{1,p}(0,1)$ ($1\leq p\leq\infty$) such that
$f>0$ a.e., and let $g = f^\alpha$ ($0<\alpha<1$).
I am wondering if
$$g'=\alpha f^{\alpha-1} f'\ (weak)\qquad a.e. ?$$
Can somebody ...
- 29
0
votes
0
answers
26
views
relation between the norm of Sobolev space $H_1$ and $L^p$ norm for non-increasing radial function
I am interested to find
$$\sup\|u\|_{p}^{p},$$
where the sup is over non-increasing radial functions $u$ on the unit ball $B_{1}$ of $\mathbb{R}^{n}$ such that
$$\|u\|_{H_1}^2 <r$$
for some $r >...
- 1
1
vote
0
answers
22
views
How to bound non-linear terms of integral operator in Sobolev Space
Let $I=[0,1]$ and consider the function $F:H^1(I)\to \mathbb R$ given by $$F(u)=\int_I \bigr( u(t) \bigr)^4 dt. $$
This makes sense, because $H^1(I)\hookrightarrow C^0(I).$ I want to understand the ...
- 1,264
0
votes
0
answers
38
views
Function in $H^1(\mathbb{R})$
Consider the function $$f=|x|^pe^{-x^2},$$ where $p$ is a real constant. The function $f$ is in $L^2(\mathbb{R})$ iff $p>-1/2$.
The function $f$ is in $H^1(\mathbb{R})$ iff $p>1/2$ or $p=0$.
I ...
0
votes
1
answer
32
views
Bessel inequality for $H^1$ function
Let $U\subset \Bbb{R}^n$ be the bounded sufficient smooth open set, let $g\in H^1(U)$, and let $\{w_k\}$ be a sequence of the orthonormal basis for $L^2(U)$ and orthogonal basis in $H^1(U)$ (which can ...
- 2,928
1
vote
1
answer
44
views
Lax-Milgran theorem with the Poisson equation
Let $ \Omega $ be a bounded domain in $\mathbb R^3$ with smooth boundary. Consider the Poisson equation
$$
-\Delta u=f
$$
where $ f\in C_0^{\infty}(\Omega) $ and $f$ is null outside $\Omega$.
I'm not ...
- 1,621
0
votes
1
answer
23
views
Sobolev embedding implies lowerbound on the volumn of the ball
Assume Sobolev embedding：
$$H_{1}^{1}(M) \subset L^{n /(n-1)}(M)$$
holds for the compact Riemannian manifold $(M^n,g)$, then we can get
$$\operatorname{Vol}_{g}\left(B_{x}(r)\right)^{(n-1) / n} \leq C ...
- 2,928
1
vote
0
answers
59
views
Mean value theorem in Sobolev spaces
Let $\Omega$ be open and sufficiently smooth, $0\leq \tau \leq T$ and consider $u \in L^2(0,T; H^1(\Omega)), \partial_t u \in L^2(0,T;L^2(\Omega))$.
Can we conclude $$\int_0^\tau||u(\tau)-u(t)||^2_{L^...
- 41
0
votes
0
answers
29
views
Gradient of a functional defined on an Hilbert space (with respect to a $W^{1,2}$ inner product)
$\newcommand{\R}{\mathbb R}$
Consider the Hilbert space $X = W^{1,2}(\R)\oplus W^{1,2}(\R)$ (Sobolev spaces).
I define a function $F:X\to \R$ as
$$F(u,g)= \int_\R u(t)\partial_tg(t) dt.$$
$F$ is ...
1
vote
0
answers
44
views
Equivalent definitions of Sobolev space on manifold and references
It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$:
D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm.
D2. All functions ...
- 1,073
1
vote
1
answer
32
views
How to deduce from these inequalities that $x_n\to 0$ in $W_0^{1, p}(\Omega)$?
Let $p, h, d\in\mathbb{R}, h>0, p>1$ and $n\in\mathbb{N}, n\ge 2$. Let $(x_n)_n\subset (W_0^{1, p}(\Omega),\|\cdot\|)$. During the class of today, the lecturer said that the inequalities
$$\|x_n\...
- 807
1
vote
1
answer
22
views
Exercise 8.19 - lower semicontinuous - l.s.c - Brezis
I'm sorry, but I don't know how to do the second part of item 1 of this exercise:
Show that $\phi$ is l.s.c, that is, show that for each $\lambda \in \mathbb{R}$ set
$$
[\phi \leq \lambda ] = \lbrace ...
- 126
0
votes
0
answers
36
views
Examples of open and bounded sets in $\mathbb{R}^d$ satisfying the uniform cone condition
Cone conditions on domains are extensively used in the proof of the Sobolev embedding theorem. Somewhat less common in that respect is the uniform cone condition, which Adams and Fournier (Sobolev ...
- 5,720
2
votes
0
answers
37
views
How to prove that normed space is complete?
$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
- 521
0
votes
0
answers
5
views
minimal induced norm on sobolev space
I am new to functional analysis and induced norms. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a challenging question for me. Could ...
2
votes
1
answer
46
views
A coercive bilinear form
Let $\alpha > 0$ and $X = [H^{2}(\Omega)\cap H^{1}_0(\Omega)] \times H^{1}_{0}(\Omega)$.
Find $\lambda_0 > 0$ for which the bilinear form $B: X \times X \rightarrow \mathbb{R}$ given by
$$
B((u,...
0
votes
0
answers
38
views
Power of Sobolev function: if $ u \in H^2(\Omega) $, will $ |u|^{1+p} \in H^s(\Omega) $ for some $ s>1 $? Assume that $ p \in (0, 1) $.
Here, $ \Omega \subset \mathbb{R}^n $ is a bounded domain which can be as nice as you want (such as $ \mathbb{T}^n $ or $ B_r(0) $). I have an $ H^2 $ funciton $ u $ and I want to know if there is ...
- 67
0
votes
1
answer
25
views
Can we use Newton-Leibnitz for $W^{1,p}$ function
For a function $u\in W^{1,p}$, we always use a sequence of $C^1$ function to approach it and derive the consequences. However, can we just claim for a.e. x, y:
$$u(x)-u(y)=\int_0^1 Du(y+t(x-y))\cdot (...
- 63
1
vote
0
answers
36
views
Interpolation of Bochner-Sobolev spaces
Does anyone know how to prove or has an explicit reference for interpolation inequalities of the form
$$ \|f\|_{H^{l}(0,T;H^{(1-l)}(\Omega))} \leq C \|f\|_{H^{1}(0,T;L^2(\Omega))}^l\|f\|_{L^2(0,T;H^1(\...
- 88
1
vote
0
answers
47
views
Composition $F\circ g$ of Sobolev function $g \in W^{k,2}(\mathbb R^n)$ with smooth map $F$ is Sobolev
$\newcommand{\R}{\mathbb R}$
Let $F: \R\to \mathbb R$ be a smooth function such that $F(0) = 0$.
Now consider $g\in W^{k,2}(\R^n)$ (Sobolev space) with $k>n/2$.
Is it true that $F\circ g\in W^{k,...
5
votes
1
answer
42
views
Remark 11 - Theorem 9.12 (Morrey) - Brezis - Pag 282 [closed]
I'm not sure about Remark 11.
(i) Why is $\mathbb{R}^{N}\setminus A$ dense in $\mathbb{R}^{N}$???
(ii) How to ensure that the ongoing representative he defined is unique?
- 126
0
votes
0
answers
10
views
Sobolev embedding for stochastic processes
Let's say I have some stochastic process $u_t \in L^2 ((0,T), H^2 (\Omega))$ with $\Omega$ as regular as you want in $\mathbb{R}^2$.
If $E\left[ \int_0 ^ t ||u_t||_{H^2} ^2 \right] \leq C$, do I ...
- 2,753
1
vote
1
answer
39
views
Reference for Sobolev-Hölder embedding for unbounded domains
I would like to know whether the following is true and references:
If $\Omega\subset\mathbb R^n$ is open (not necessarily bounded) and $k = n/p + r+\alpha$, and $\alpha \in (0,1), r\in \mathbb N, k\...
0
votes
0
answers
39
views
About a remark in Evans PDE, in the regularity of elliptic equation
On Page 341, after Theorem 5 (Higher boundary regularity) in Chapter 6.3 (Regularity), Evans states that
Remark. If $u$ is the unique solution of $$
\left\{\begin{aligned}
L u=f & \text { in } U \...
- 23
4
votes
0
answers
37
views
Derivative of a map $f:\mathbb R \to \ell^2 $ to a separable Hilbert space vs derivative of each component of the Hilbert basis.
${\newcommand{\R}{\mathbb{R}}}$ Let $\ell^2 $ be the Hilbert space of square summable sequences of real numbers.
Consider a map $f: \R \to \ell^2$, that has components $f_n:\R\to \R$, i.e. $f(t)=\{...
0
votes
1
answer
37
views
About the continuity of the integral on the boundary of a ball of a $H^1$ function
I’m considering a $H^1$ function u on a open domain D. Is the integral:
$$ \int_{\partial B_r(x)} u \hspace{2pt}dH^{n-1}$$
continuous with respect to x?
I tried to prove that it’s differential by ...
- 63
2
votes
0
answers
62
views
Assumptions in Schauder Fixed Point Theorem
I have a - maybe slightly stupid - question about the Schauder-Fixed-Point Theorem.
The formulation I have in mind is:
Let $A$ be a closed, convex, nonempty subset in a Banach space $(X,\|\cdot\|)$, ...
- 158
0
votes
1
answer
42
views
Density of $[H^2(\Omega) \cap H^1_0(\Omega)] \times H^1_0(\Omega)$ in $H^1_0(\Omega) \times L^2(\Omega)$
I am reading Evans' PDE proof of theorem 6, section 7.4.3, page 444 where the following is said to be "clear"
$[H^2(\Omega) \cap H^1_0(\Omega)] \times H^1_0(\Omega)$ is dense in
$H^1_0(\...
- 479
0
votes
1
answer
19
views
Prove a certain functional is unbounded from above
Let $\lambda\in\mathbb{R}$, $2<p\leq 2^*=(2N)/(N-2)$. For $u \in H^{1}_0({\Omega})$ where $\Omega$ is a domain of $R^N$. Define:
$$ \varphi(u) = \displaystyle\int_{\Omega} \frac{1}{2}|\nabla u|^2 +...
- 179
0
votes
2
answers
37
views
The relations between $ L^2 $ and $ H^{-1} $.
Let $ \Omega $ be a bounded domain in $ \mathbb{R}^d $. Suppose that $ f\in L^2(\Omega) $. Define $ \phi\in H^{-1}(\Omega) $ such that
\begin{align}
\langle\phi,u\rangle_{H^{-1}\times H_0^1}=\int_{\...
0
votes
0
answers
22
views
If $\|(x_n, y_n)\|_W\to +\infty\quad\mbox{ as } n\to +\infty,$ we can assume WLOG that $\|x_n\|_{W^{1, p}}\to +\infty$?
Let $\Omega$ be an open bounded domain and let $p, q>1$. Consider $(x_n)_n\subset W^{1, p}(\Omega)$ and $(y_n)_n\subset W^{1, q}(\Omega)$.
Defined
$$W= W^{1, p}(\Omega)\times W^{1, q}(\Omega)\quad\...
- 807
0
votes
0
answers
14
views
Reproducing kernel hilbert spaces: variation norm in arbitrary dimensions
I am currently dealing with the topic of reproducing kernel Hilbert spaces (RKHS) given the draft book of Francis Bach.
As a background knowledge for my current problem define:
\begin{align}
&H_1=\...
- 53
1
vote
0
answers
23
views
If $(u, v)\in W^{1, p}(\Omega)\times W^{1, q}(\Omega)$, what does the notation $|(u, v)|$ mean?
Let $\Omega$ be an open bounded domain, $p, q>1$ and let $W= W^{1, p}(\Omega)\times W^{1, q}(\Omega)$. Let $(u, v)\in W$.
What does the notation $|(u, v)|$ mean?
It is in my math class notes, but. ...
- 2,345