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.
5,524
questions
2
votes
1
answer
106
views
How to interpret $|\nabla u|$ when using distributions?
In Sobolev spaces, to define the respective norm, we need to be able to interpret $||\nabla u||_{L^2(\Omega)}=\left( \int_{\Omega} |\nabla u|^2 \ dt\right)^{1/2}$
However, how do we interpret $|\...
- 5,251
0
votes
1
answer
119
views
The norm of a distribution (generalized function) in a Sobolev Spaces involving time.
Being a distribution a linear continuous functional in the test function space $\mathcal{D}(0,t)$ (for the purpose of my question), and $$||u ||_{L^p(0,T;V)}:= \left(\int^T_0||u||^p_V \ dt\right)^{1/p}...
- 5,251
2
votes
0
answers
74
views
Existence of trace operator in$ W^{s,p} (D)$
Let $D$ be a smooth, bounded domain. $W^{s,p} (D)$ denotes the fractional Sobolev space defined as in Hitchhiker's Guide to Fractional Sobolev Space. Assume that $s>\frac{1}{p}$. I want to show ...
- 491
0
votes
1
answer
33
views
Deducing this inequality for $u\in H^1(a,b)$
In the following picture, $v,u \in H^1(a,b)$ with $(a,b)\subset \mathbb{R}$.
Well, I think there is a typo.
How do we get $$u(x)+\int^y_x u'(s)ds\leq u(x) +\sqrt{b-a} ||u'||_{L^2(a,b)}\ ?$$
I'm ...
- 5,251
1
vote
0
answers
35
views
Proof of an inequality involving $Q\in H^{1}(\mathbb{R}^d).$
I am trying to understand a special case of Diamagnetic inequality mentioned in Appendix [B] of the book Nonlinear Dispersive Equations by Tao. Here is the statement,
Let $Q\in H_x^{1}(\mathbb{R}^d)...
- 9,196
1
vote
0
answers
81
views
Reference request: definition of Sobolev space on manifolds using positive elliptic operator
I've been reading a paper "Schatten classes on compact manifolds: Kernel conditions" by Delgado and Ruzhansky (https://arxiv.org/abs/1403.6158).
In the paper, the authors define the Sobolev space $H^\...
- 531
1
vote
0
answers
103
views
$\Delta u=\sin(u)$ implies $u$ smooth for weak solution $u\in H^1_0(\Omega)$
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, $\partial\Omega\in C^\infty$. Let $\Delta$ be the standard Laplacian. Suppose $u\in H^1_0(\Omega)$ is a weak solution to $$\begin{cases}\Delta u=\...
- 8,556
5
votes
1
answer
2k
views
Understanding a Sobolev Embedding Theorem
In my adv. Analysis course, we have studied the following Sobolev Embedding Theorem:
Let $m\in\mathbb{N}$ and $s>m+d/2$. Then $$H^s(\mathbb{R}^d)\hookrightarrow C_0^m(\mathbb{R}^d)$$That is: $H^...
- 2,601
1
vote
1
answer
89
views
Equivalence of norms in $L_p$ spaces.
Let $(f,g) \in {W_0}^{1,p}(0,1) \times {L^p}(0,1), p\in[1,\infty]$. I want to prove the equivalence of the two norms $${\left\| {{{{f_x} - g} \over 2}} \right\|_p} + {\left\| {{{{f_x} + g} \over 2}} \...
- 1,523
11
votes
2
answers
6k
views
How to think of homogeneous Sobolev spaces
The homogeneous Sobolev space $\dot H^s(\mathbb{R}^d)$ can be defined as the completion of $\mathscr{S}(\mathbb{R}^d)$ (the space of Schwartz functions) under the norm
$$
\|f\|_{\dot H^s(\mathbb{R}^d)...
- 5,106
1
vote
1
answer
76
views
Liouville's theorem for conformal mappings: differentiable functions?
Apparently Liouville's theorem for conformal mappings holds for functions in $W^{1,n}$. Am I to understand that potentially, there are everywhere differentiable functions to which it doesn't apply?
- 6,043
1
vote
1
answer
225
views
Does this inclusion for Besov function spaces hold and how to prove it?
Consider the Besov spaces $B_{p,q} ^s$ consisting of those functions $f$ such that
$$f \in W^{n,p} (\mathbb{R}), \quad \int_{0} ^{\infty} \Big | \frac{\omega_p ^2(f^{(n)}, t)}{t^{a}} \Big |^q \frac{...
- 417
2
votes
1
answer
449
views
Hardy-Littlewood-Sobolev inequality using generalised Young inequality in Lorenz spaces
I want to prove that
$$\left| \left| \frac{1}{|x|^a} \ast f \right| \right|_q \lesssim ||f||_p$$
with $1 < p < q < \infty$ and $a= n \left(1 + \frac{1}{q}- \frac{1}{p} \right)$ using a ...
- 3,354
0
votes
1
answer
101
views
Is a $W^{1,\infty}$ function a continuous function?
Let consider a function $f\in W^{1,\infty}([a,b];\mathbb{R}^n)$.
Somebody can suggest me a reference where I could confirm if $f$ is a continuous function, due to $f \in W^{1,\infty}([a,b];\mathbb{R}^...
- 63
5
votes
0
answers
327
views
Equivalent norms on Sobolev spaces. Only function and last derivative needed to define the norm.
Equip $W_n^{p}[0,1]$ with the norm
$$\left\|f\right\|_{W_n^{p}} = \sum_{k=0}^{n}\left\|f^{(k)}\right\|_{L^p}.$$
I want to prove that this norm is equivalent to the norm
$$\left\|f\right\|_2 = \left\|...
- 1,381
0
votes
0
answers
65
views
How to find a constant such that this inequality holds in $H^1$
Let $a$, $b \in \mathbb{R}$. Does there exist a $0 < c \in \mathbb{R}$ such that $c(u(a)^2+u(b)^2) \leq \left\lVert u \right\rVert_{H^1}^2$ for all $u \in H^1(a,b)$?
I can't find an explicit bound ...
- 33
0
votes
1
answer
462
views
Convergence of Lipschitz functions in $L^\infty$ implies the convergence in $H^{1}$
Suppose $f_n \to f$ in $L^\infty(U)$ where $f_n, f \in C^{1,1}(\bar{U})$, (Holder space with $k=1, \alpha =1$, i.e, Lipschitz functions) where $U$ is an open and bounded in $\mathbb{R}^n$.
Q: Can we ...
- 1,944
1
vote
1
answer
56
views
Show $u \notin H^2$ for $-\text{div}(a\Delta u)=1 $ where $a$ is a step function
I am working on a problem which shows interior regularity may fail if you have non-Lipschitz coefficients.
Let $\Omega = [−1,1]$ and $K = [−1/2,1/2] \in \Omega$. Define $a(x)=1+1_K(x)$ and ...
- 1,681
1
vote
0
answers
375
views
Weak solution of Laplace equation in $H^2$
Let $\Omega = [0,1]$ and $f\in L^2(\Omega)$. Prove that the weak solution $u \in H_0^1(\Omega)$ of the Laplace
equation with homogeneous Dirichlet boundary conditions $$-u''=f \text{ in } \Omega, \ ...
- 1,681
1
vote
0
answers
24
views
Weighted integrability of Sobolev functions in $H^2_p$
Suppose $u \in H^2_p(\mathbb{R}^n)$. What can be said about the decay of $u$ in an $L^p(\mathbb{R}^n)$ sense, i.e. is it true that
$$\int_{\mathbb{R}^n} |xu(x)|^pdx < \infty?$$
- 179
0
votes
1
answer
52
views
Convolution of (s+1) characteristic functions
I think it is possible that if I will be able to show that g is in $L^2(\mathbb{R})$ and show that the integral of $|\hat{g}(\omega)|^2(1+\omega)^s d\omega$ over $\mathbb{R}$ is finite, then it is in ...
- 21
0
votes
1
answer
65
views
permuting two integrations
Can someone explain to me what he used after he said 'and permuting the two integrations with respect t and x we conclude ..'
Lemma Let $\Omega$ be an open set of $\mathbb{R}^{N}$ bounded in at least ...
- 299
4
votes
1
answer
685
views
An equivalent norm in a subspace of $H^2 (\Omega)$
The following questions concerns a problem I am treating in my Masters dissertation.
Let $\Omega $ be an open, bounded domain in $\mathbb{R}^3$. Then the norm
$$
\Vert u\Vert^2 = \Vert u\Vert_2^2 + ...
- 3,046
0
votes
1
answer
63
views
If $\,-u''+u=f$, then $\,\|u\|_{L^s}+\|u'\|_{L^q}+\|u''\|_{L^p}\le c\|f\|_{L^p}$
Let $u\in \mathcal{S}(\mathbb{R})$ (Schwartz space) be a solution of the equation
$$
-u''+u=f, \quad \text{where}\,\,\,f \in \mathcal{S}(\mathbb{R}).
$$
Show that for all $1\leq p,q,s \leq \...
- 1,681
0
votes
1
answer
43
views
Weakly equal functions and its decomposition
Suppose $f - g \in V$ where $V$ is a Hilbert space.
If $f, g \in V$ and
$$
\langle f-g, v\rangle_V = 0, \forall v \in V^*,
$$
we say $f$ and $g$ are weakly equal in $V$.
I am wondering if there is a ...
- 1,944
2
votes
2
answers
443
views
Strong $L^2$ convergence to zero implies weak convergence in $H_0^1$?
Suppose $f_k \to 0$ in $L^2(U)$ where $U$ is an open and bounded domain in $\mathbb{R}^n$.
Suppose further that $f_k \in H_0^1(U)$ where $H_0^1(U)$ is the Sobolev space whose trace being zeros.
I ...
- 1,944
4
votes
2
answers
713
views
Find a weak differentiable function which is discontinuous
I'm seeking a weak differentiable function $f \in W^{1,1}(\Omega)$ which is discontinuous.
I think every weak differentiable function defined on $\mathbb{R}$ is continuous by
Sobolev embedding ...
- 41
1
vote
1
answer
382
views
Local vs global Sobolev space
Let
\begin{equation}
H_{\text{loc}}^{2}(\mathbb{R}^d)=\{u:\mathbb{R}^d\to\mathbb{R}\;|\;u\in H^{2}(V)\text{ for all }V\subset\subset \mathbb{R}^d\}.
\end{equation}
I was wondering if it is true that
\...
- 697
2
votes
1
answer
55
views
Integral of $f(x,y) \in W^{1,1}(B)$
I have the follow problem. Let $B \subset\Bbb R^2 $ the unitary open ball, and let
$$
f(x,y):=
\begin{cases}
1+x^2+y^2 & \text{ if }x>0,\\
a(y-1)^2 +b(y+x^2) & \text{ if }x<0,
\end{...
- 500
2
votes
0
answers
132
views
Question about the definition of Bessel potentials $H^{s, p}(\mathbb{R}^n)$ and their embeddings
Usually the Bessel potential spaces are defined as
$$
H^{s, p}(\mathbb{R}^n)=\{u\in\mathscr{S}^{\prime}(\mathbb{R}^n):\mathcal{F}^{-1}(\langle\cdot\rangle^s\hat{u})\in L^p(\mathbb{R}^n)\}
$$
only for ...
- 508
3
votes
1
answer
59
views
If $u_n \rightharpoonup u$ in $H_0^1 (\Omega)$ then $u_n^2 \to u^2$ in $L^{6/5}(\Omega)$
My question regards a minor detail in the proof of a lemma in a research paper I am reading.
Let $\Omega$ be an smooth, bounded open set of $\mathbb{R}^3$ and suppose $u_n \rightharpoonup u$ in $H_0^...
- 3,046
1
vote
1
answer
66
views
Does $\|f\|_{L^2}\,\le\,C(\|f'\|_{L^2} + \|xf\|_{L^2})$ for $f\in H^1(-1,1)$?
I would like to prove that there is a constant $C>0$ such that for all $f\in H^1(-1,1)$ (the usual Sobolev space) we have
$$
\|f\|_{L^2}\,\le\,C(\|f'\|_{L^2} + \|xf\|_{L^2}).
$$
Here, $xf$ is just ...
- 10.6k
3
votes
1
answer
402
views
Trace operator in bounded domain that is not Lipschitz
I know that Trace operator $T : W^{1,p}(\Omega) \to L^p(\partial \Omega, H^{n-1})$ is well defined when $\Omega$ is a bounded Lipschitz domain. (here $H^n$ is the Hausdorff measure). What about only ...
- 1,068
2
votes
1
answer
191
views
A funny exercise with bounded Lipschitz domain and Sobolev spaces
I'm in trouble with this exercise
Let $\Omega \subset \mathbb{R^n}$ be a bounded Lipschitz domain and let $p \in ]1,n[$. prove that there exists $C>0$ such that, for every $f \in W^{1,p}(\Omega)$...
- 1,068
2
votes
0
answers
268
views
Density of $C^{\infty}(I) \cap W^{1,\infty}(I)$ in $W^{1,\infty}(I)$
I was inspired by this question and Meyers-Serrin theorem to prove that there exists a function $u \in W^{1,\infty}(I) $ such that there are no functions $u_m \in C^{\infty}(I) \cap W^{1,\infty}(I)$ ...
- 1,068
0
votes
0
answers
332
views
Variational formulation Poisson equation (1d FEM)
I have the same question of this one
$\bullet $
In the answer I've seen that a "Lifting" function is used, is it to have a formulation where the test functions are in the same space of the solution?
...
- 1,603
1
vote
0
answers
114
views
About extension on Sobolev space
Let $\Omega\subset\mathbb{R}^n$ be open connected with smooth boundary. Let $\zeta$ be a zero extension operator : $\forall u\in W^{k,p}(\Omega)$,
$$\zeta u := \Big\{\begin{array}uu \quad\text{on $\...
- 1,635
1
vote
0
answers
50
views
Fixed Point Theorem in Topological vector spaces
Suppose we have a decreasing family of Banach spaces $H^s$ (the prototype is some variant of the Sobolev space $H^s=W^{s,2}$, so you can assume that they are Hilbert spaces if you want), and we ...
- 65
0
votes
1
answer
42
views
A density problem in Sobolev space?
For $ k,p \in \mathbb{N} $, denote $ k $ order classical Sobolev space on $ (0,2\pi) $ by $ H^k(0,2\pi) $ and define
\begin{equation*}
\mathcal{H}^{k+p}_0:= \{ \varphi \in H^{k+p}(0,2\pi): \varphi(0) =...
- 411
1
vote
1
answer
145
views
Weak limit of gradient of functions
Consider $\{f_n\}\subset H^1(\Omega)$, where $\Omega\subset\mathbb{R}^2$ is an open bounded set. Assume we have $\|\nabla f_n\|_{L^2(\Omega)}\le C$ for a constant $C$ uniformly. Then, we can conclude ...
- 670
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votes
1
answer
68
views
$u \in W^{1,p} (U)$ Implies $u^{+}, u^{-} \in W^{1,p}(U)$ for $U \subseteq \mathbb{R}^N$ Bounded (Evans, Chapter 5, Exercise 18)
Full Problem
Assume $1 \leq p \leq \infty$, $U \subseteq \mathbb{R}^N$ bounded. Show that $u \in W^{1,p} (U)$ implies $u^{+}, u^{-} \in W^{1,p}(U)$, and
$Du^+ = \begin{cases} Du, \ \text{a.e. on } \...
- 457
2
votes
0
answers
331
views
Compact Embeddings
Let $\Omega$ be a bounded open with no assumptions on the boundary $\partial\Omega$. Show that $H^{k+1}_0(\Omega)$ is compactly embedded in $W^{k,2}(\Omega)$.
Show that if $\Omega$ is bounded, then $...
- 541
0
votes
1
answer
31
views
Is $\left\{ u\in H^{1}\left(\Omega\right)\left|\,a<u\left(x\right) <b\:\text{a.e.}\right.\right\}$ open in $H^1(\Omega)$
Let $\Omega$ bounded, open, connected and Lip. domain.Let positive constants $a,b$ such that $a<b$.
Is $A = \left\{ u\in H^{1}\left(\Omega\right)\left|\,a<u\left(x\right)
<b\:\text{a.e.}\...
- 85
4
votes
2
answers
108
views
Exact solution $\int_0^1 u'v'=v(1/2)$
This question concerns a variational form of the Laplace equation with homogeneous Dirichlet boundary conditions: $$-u''=f \text{ on } [0,1], u(0)=u(1)=0.$$
Let $V=H^1_0(\Omega), \Omega=[0,1]$ and $...
- 1,681
1
vote
0
answers
241
views
Generalized definition of the Green Formula (In Sobolev Spaces)
Given the classical version of Green's Formula:
Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$ and $u, v \in C^{2}(\overline{\Omega})$.
\begin{equation}
\int_{\Omega} Dv \cdot Du \ dx = -...
- 73
0
votes
1
answer
191
views
is $H^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)$ closed in $H^{1}\left(\Omega\right)$?
Let $\Omega$ bounded, open, connected and Lip. domain. Is $H^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)$ closed in $H^{1}\left(\Omega\right)$? i.e., with the norm of $H^{1}\left(\Omega\...
- 85
1
vote
1
answer
87
views
If $\int_U |Du|^2 dx + \int_{\partial U}u^2 dx < \infty$ is $\int_U |u|^2 dx < \infty$?
Do we have
$\int_U |Du|^2 dx + \int_{\partial U}u^2 dx < \infty \implies \int_U |u|^2 dx < \infty$ when $u \in H^1(U)$ and $\partial U $ is smooth ?
This result is true for $u \in H_0^1(U)$ ...
- 3,376
0
votes
1
answer
33
views
$u_n\rightarrow u$ and $\nabla u_n\rightarrow g$ implies $g=\nabla u$?
Suppose that $u_n\in H^1(\Omega)$ and $u_n\rightarrow u$ in $L^2(\Omega)$ and one shows that $\nabla u_n$ has a limit in $L^2(\Omega)$, let's say $\nabla u_n\rightarrow g$. I wonder if $g=\nabla u$. ...
- 1,233
4
votes
0
answers
77
views
How to make sense of Sobolev spaces on bounded domain from the point of view of distributions.
In the theory of $\mathbb{R}^{d}$, one defines the space of Schwartz functions $ \mathcal{S}(\mathbb{R}^{d}) $ to be the space of smooth functions decaying faster than any polynomials. However, when ...
- 693
0
votes
0
answers
238
views
A little help to show strong convergence in Sobolev space $H^1$
Let $u_n $ be a sequence in $H^1 (\Omega,\mathbb{C})$ where $\Omega \subset \mathbb{R} ^N$ is bounded. Assume that we know that all the functions $u_n$ are smooth and $\Vert u_n \Vert _{C^m} \leq K(m,...
- 1,233