Sobolev spaces are function spaces generalising the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, functions in a Sobolev space have also differentiability conditions: that is, we require all partial derivatives of the function up to a ...
2
votes
2answers
215 views
Help with Evans PDE problem
I have trouble proving the following problem (Evans PDE textbook 5.10. #15). Could anyone kindly help me solving the problem? I know that I should somehow use Poincare inequality but I still cannot ...
2
votes
1answer
40 views
How to esimate $\inf\int|\nabla g|^p\,dx$
It is rather easy question but I'm already struggling with this problem for a long time.
I'm trying to estimate the value
$$\inf\int|\nabla g|^p\,dx$$
where $\mathbf{inf}$ is taken over all ...
1
vote
1answer
67 views
Sobolev spaces of infinite order
I do have a question about the Sobolev spaces of infinite order. Let me first define them:
Let $H^s(\mathbb{R}^n)$ denote the Sobolev space of order $s \in \mathbb{R}$. We can naturally identify ...
3
votes
1answer
74 views
Sobolov Space $W^{2,2}\cap W^{1,2}_0$ norm equivalence
I would like to know why on $W^{2,2}\cap W^{1,2}_0$ the norms
$$ ||u|| _{W^{2,2}}=\sum_{|\alpha|\leq 2}||D^\alpha u||_{L^2}$$
and
$$||\Delta u||_{L^2}$$
are equivalent.
2
votes
1answer
166 views
Dual space of the sobolev spaces.
What is the dual space of $ H¹(\Omega) = W^{1,2}(\Omega) $? What is the dual space of $ W^{m,p}(\Omega) $? I know for example that the dual space of $ L^{p}(\Omega) $ for $ 1 \le p < \infty $ is $ ...
4
votes
1answer
221 views
Show that norm of a functional is continuous
This question is based on Lemma 3.3, page 6 in this paper: http://arxiv.org/pdf/1106.0622v4.pdf
I changed the notation quite a lot, but it should be a one-to-one correspondence.
$S(x)$ is a compact ...
1
vote
1answer
101 views
Vanishing at the infinity of a function in the Sobolev space.
If $f \in H^s (\Bbb R^n)$ for $s > 1 + \frac{n}{2} $ then the Sobolev inequality implies that $f$ and $\nabla^\alpha f$ ($|\alpha| =1$) vanishes at the infinity. ($\alpha$ : multi-index). But in ...
0
votes
0answers
82 views
Integrating $\ln(1+|\ln|x||)$ in $B_1(0)$
I am trying to integrate $$\int_{B_1(0)} \ln(1+|\ln|x||).$$ $B_1(0)\subset \mathbb R^n$
Basically what I am trying to see is whether $\ln(1+|\ln|x||)$ belongs to $L^\infty (B_1(0))$ and ...
1
vote
1answer
137 views
On compact embedding of Sobolev space?
Let$\{u_k\}\in W^{1,p}(R^n)$, $p\in[1,n)$, $\sup_k||u_k||_{W^{1,p}(R^n)}<C$, then $\forall r>0$ and $\forall q\in[1,p^*)$,where $p^*=\frac{np}{n-p}$, there exist a subsequence $\{u_{k_i}\}$ of ...
0
votes
0answers
55 views
Compact embedding theorem of $W^{k,p}(R^n)$?
Is there some kind of function space $X(R^n)$ which satisfies the compact embedding relation as follows: $W^{k,p}(R^n)\hookrightarrow\hookrightarrow X(R^n)$? Could I guess the indeterminate function ...
1
vote
0answers
100 views
When the weak derivative just is the strong (or classical) derivative?
When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
1
vote
1answer
125 views
Infimum of the $L^2$ norm of the derivative of functions of $H^1_0$ of $L^2$ norm $1$
Let $I=(a,b)$ with $a<b$ real numbers. Define $\lambda_0$ by $$\lambda_0=\inf\{\|u'\|_{L^2(I)}^2:\ u\in H_0^1(I),\ \|u\|_{L^2(I)}=1 \}.$$
How can I prove that ...
2
votes
1answer
153 views
Counterexample of Sobolev Embedding Theorem?
Is there a counterexample of Sobolev Embedding Theorem? More precisely, please help me construct a sobolev function $u\in W^{1,p}(R^n),\,p\in[1,n)$ such that $u\notin L^q(R^n)$, where ...
2
votes
2answers
118 views
Density in sobolev spaces
Is $H^{s+1} (\Bbb R^n)$ dense in $H^s(\Bbb R^n)$ for $s = 0,1,2, \cdots$ ? ($H^s$ : general sobolev space)
1
vote
0answers
28 views
One of conjectures of De Giorgi
conjecture: If $\exp(tw), \exp(tw^{-1}) \in L^1_{\operatorname{loc}}$ for each $t > 0$ then $w$ is regular weight.
$w$ is regular if weighted Sobolev space $W^l_p(\Omega,w)$ is equal to the ...
2
votes
0answers
57 views
Weak derivative and homeomorphisms commute
Suppose $\phi:H^1(S) \to H^2(R)$ is a linear homeomorphism, where $S$ and $R$ are compact surfaces in $\mathbb{R}^n$.
Let $\varphi \in D(0,T;H^1(S))$ be a $H^1(S)$-valued $C_c^\infty(0,T)$ function. ...
1
vote
1answer
54 views
Inverse estimate of gradient of Sobolev function
I need an estimate for $\| \nabla w\|_{L^2{(\Omega \subset \mathbb{R}^n)}}$, such that it is $< c\| w\|,\ w \in H_0^1(\Omega)\ $. Is this possible?
5
votes
1answer
144 views
Why no trace operator in $L^2$?
We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe.
I'm sure either ...
4
votes
1answer
103 views
Prove this inequality from functional analysis
I want to prove this equality used in out lecture notes:
Let $D=(0,r)^2 \subset \mathbb{R}^2, r\geqslant 0$. Then, for any $u \in H^1(D)$, there holds
$$\lVert u\rVert \leqslant \frac 1 r ...
4
votes
4answers
215 views
Survey papers for PDE?
I want to know if there is a good website which allows you to download survey papers on PDEs? The "survey" should include a summary of methods, skills, developments etc. I wish to get some basic (or ...
0
votes
1answer
93 views
What is the definition of smooth domains?
What is the definition of smooth domains? Is there is intuitive understanding?
3
votes
1answer
81 views
What is the use of $H_s$ for non-integer $s$?
So we have the whole set of theory for Sobolev spaces \begin{equation}
H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\},
\end{equation} and we know that ...
2
votes
1answer
61 views
How can i prove this Convergence?
Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth domain. Let $p\in (1,\infty)$ and $K=\{u\in W^{1,p}(\Omega):\ u-g\in W^{1,p}_0(\Omega)\}$, where $g\in W^{1,p}(\Omega)$. Let $\epsilon\geq 0$ and ...
1
vote
0answers
78 views
A weak convergence in Sobolev space.
Let $u \in C^0([0,T], H^{s-1}(\Bbb R^n)). $ Let $\{t_n \} \subset [0,T]$ such that $\lim_{n \to \infty} t_n = t_0$. Let $ u(t_n ) \to u(t_0) $ in the Sobolev space $H^{s-1} ( \Bbb R^n )$ for $s = ...
2
votes
1answer
65 views
$W^{k,2}(\mathbb{R}^n) \hookrightarrow L^\infty(\mathbb{R}^n)$
Let $k,n \in \mathbb{N}$ such that $n>\frac{k}{2}$. How can one prove that $u\in W^{k,2}(\mathbb{R}^n)$ may be embedded into $L^\infty(\mathbb{R}^n)$?
2
votes
1answer
106 views
A counter example to this statement
I am struggling with Evans PDE problem 5.10. #12. Here is the question: Let $V\subset \subset U$ be open sets. Show by example that if we have $\|D^hu\|_{L^1(V)}\le C$ for all $0< ...
3
votes
1answer
68 views
Invariance under diffeomorphisms of the Sobolev $H^s$ spaces
Let $\Omega, \Omega' \subset \mathbb{R}^n$ be two open subsets and let $\psi : \Omega \rightarrow \Omega'$ be a $C^k$ diffeomorphism. Then, $\psi$ induces by pullback a linear isomorphism
$$u \mapsto ...
0
votes
0answers
59 views
Good books on the relationship between function spaces $C^m([0,T],H^k(\Omega))$ and $C^r([0,T]\times\Omega)$ in details?
Good books on the relationship between function spaces $C^m([0,T],H^k(\Omega))$ and $C^r([0,T]\times\Omega)$ in details? I realy want to learn some knowledge on this kind of function spaces, but I ...
4
votes
1answer
78 views
An inequality for $W^{k,p}$ norms
Let $u \in W_0^{2,p}(\Omega)$, for $\Omega$ a bounded subset of $\mathbb R^n$. I am trying to obtain the bound
$$\|Du\|_p \leq \epsilon \|D^2 u\|_p + C_\epsilon \|u\|_p$$
for any $\epsilon > 0$ ...
2
votes
1answer
117 views
$\delta_0$ in Sobolev spaces
I'm reading a one proof. This say
If $u$ is a test function (smooth function with compact support), then
$$|\delta_0(u)|=|u(0)|=\left|\int_{—1}^0u'(t)dt\right|\leq \lVert u'\rVert_p\leq \lVert ...
2
votes
1answer
75 views
Space Sobolev $W^{m,p}$ complete
Show that Sobolev space is complete. I am trying
Than $L^p(\Omega)$ is complete then If $f_n \in L^p(\Omega)$ then $f_n \to f \in L^p(\Omega)$. But rest show that $D^{\alpha}f \in L^p(\Omega)$. How I ...
4
votes
2answers
145 views
Sobolev spaces fourier norm equivalence
I am reading about Sobolev spaces and I have a question regarding Sobolev spaces and the Fourier transform.
So by defining the Fourier transform, $F(\cdot)$ as an isometry we get ...
2
votes
2answers
77 views
Equivalent norms in the subspace of Sobolev space containing only functions with zero integral
In a problem I ended up with the vector space $V=\{f\in H^1(\Omega):\int_\Omega f=0\}$. I think it can be proven (and it would be really helpful) that $||f||_{L^2(\Omega)}\le c ||\nabla ...
6
votes
1answer
32 views
Prove that energy functional cannot be minimized
As an exercise I want to show that for
$$E(v)=\int_0^1 \frac 1 2 x^5 (v'(x))^2 - v(x) dx $$
and $H_0^1(0,1) = \{ v \in H^1(0,1) : v(0)=v(1)=0 \}$ there exists no solution to the problem ...
1
vote
1answer
53 views
Show $W^{1,q}_0(-1,1)\subset C([-1,1])$
I need show that space $W^{1,q}_0(-1,1)$ is a subset of $C([-1,1])$ space. How I will able to doing this?
3
votes
1answer
78 views
An elementary question on Sobolev space
I have a question on Sobolev space.
This is one of exercises in Evans PDE textbook.
Let $U=\{(x,y) | |x|<1, |y<1|\} \subset \mathbb{R}^2$.
Define a function $u(x,y)$ by
$$
u(x,y)=\begin{cases}
...
2
votes
2answers
156 views
How to use the dense theorem to prove this exercise?
Let $\frac{2n}{n+1}\leqslant p<n,\quad q=\frac{np}{2n-p},\quad u\in L^1(R^n)\cap W^{1,p}(R^n)$, then prove $u^2\in W^{1,q}(R^n).$
I hope someone can show me how to prove it by dense theorem ...
3
votes
1answer
243 views
Density of space in a Sobolev space
An exercise from Gilbarg-Trudinger Elliptic Partial Differential Equations states the following :
"Using Lemma 9.12, show that for a $C^{1,1}$ domain $\Omega$ the subspace $$\{u \in ...
1
vote
1answer
42 views
Given a “$L^2$ vector field”, is there a function whose weak gradient is this vector field?
Let $\Omega\subset\mathbb{R}^n$ be open and connected. Given any function $V\in L^2(\Omega,\mathbb{R}^n)=:H$, is there an $u\in W^{1,2}(\Omega)$ such that $\nabla u = V$, i.e. $\|\nabla u-V\|_H = 0$? ...
0
votes
1answer
24 views
Point evaluation of $H^{k,\infty}$-functions
I'm trying to understand a step in a proof. I don't know which of the prerequisites are required or even helpful for what I need, so I give you the complete situation:
Let $T \subset \mathbb{R}^n$ be ...
2
votes
1answer
189 views
Example of a discontinuous and bounded function for the limiting case $W^{1,n}$
Let $\Omega = B(0,1)$ be the open unit disc in $\mathbb{R}^2$. I'm looking for an example of a discontinuous and bounded function in $W^{1,2}(\Omega)$.
I know the example $u(x) = \log \left( \log ...
2
votes
1answer
38 views
Linear subspace of Sobolev space
Consider two Sobolev spaces $H^s(\mathbb R^n)$ and $H^t(\mathbb R^n)$ where $s>t$, if $V$ is a linear subspace of $H^s(\mathbb R^n)$ such that there exists a constant $C$ and any $f \in V$, we have ...
0
votes
0answers
49 views
Time dependent or Bochner space references?
Does anyone have any recommendations where I can learn about time dependent or Bochner spaces? I mean spaces like $L^p(0,T; H^{-1}(\Omega))$. I think one needs some knowledge of distributions, so any ...
12
votes
1answer
265 views
$\tilde{u}=0,\ a.e.\ x\in\Gamma$?
Suppose $\Omega\subset\mathbb{R}^n$ is a bounded domain. Let $u\in H_0^1(\Omega)$ and $\tilde{u}\in L^2(\Omega)$ satisfies $$\int_\Omega\nabla u\nabla v=\int_\Omega \tilde{u}v,\ \forall\ v\in ...
1
vote
1answer
89 views
Dual space norms and equivalence
Suppose $S(r)$ is set parametrised by $r \in [0,T]$. Let $\phi_t^s : H^1(S(t)) \to H^1(S(s))$ is a linear homeomorphism.
Suppose $\lVert \cdot \rVert_{H^1(S(t))}$ and $\lVert \phi_t^s(\cdot) ...
2
votes
2answers
63 views
Sobolev Spaces and Measure Theory
Suppose $u\in W^{1,p}_0(\Omega)$, where $\Omega\subset\mathbb{R}^n$ is a bounded domain (open and connected) and $p\geq 1$. Let $a\in\mathbb{R}$ and suppose the set $\Omega_a=\{x\in\Omega:\ u(x)=a\}$ ...
1
vote
1answer
54 views
Can we obtain the following variation of Poincaré inequality?
The known Poincaré inequality says that in the conditions of the theorem we have
\begin{equation}
\|u - u_{\Omega}\|_{L^{p}(\Omega)} \le C \| \nabla u \|_{L^{p}(\Omega)}.
\end{equation}
see for ...
6
votes
0answers
120 views
Regularity of elliptic PDE with coefficients in some Sobolev space
Is there a regularity theory for elliptic equations "optimized" for coefficients in a Sobolev space $W^{k,r}$?
By this, I mean a result (and the according elliptic estimates) that gives you (for $k$ ...
0
votes
0answers
48 views
Proving $ f \in C_b^1 ( [0,\infty) \times \Bbb R^n) $ by using the Sobolev inequality.
Let $s > 1 + n/2$ for $n \in \Bbb N$, and $s$ be an integer. If $f \in C^0 ( [0,\infty), W^{s,2} (\Bbb R^n )) \cap C^1 ( [0, \infty) , W^{s-1,2}(\Bbb R^n)) $ then how can I show that $$ f \in C_b^1 ...
1
vote
1answer
50 views
Identify the distrionbutional derivative with classical derivative?
I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma.
In proving the theorem, he defines the function $F$, and calculates its ...
