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 ...
-1
votes
1answer
52 views
Show reflexivity of Sobolevspace $W^{1,4}(0,1)$
I would like to show elementary - using the canonical embedding - that the Sobolevspace $W^{1,4}(0,1)$ is reflexive.
Therefore I set $X=W^{1,4}(0,1)$ and now I have to show that the canonical ...
2
votes
1answer
140 views
Compact Embedding of $W^{1,2}(0,T;\mathbb{R}^d)$ in $C(0,T;\mathbb{R}^d)$
I need to prove (if true) that the space $W^{1,2}(0,T;\mathbb{R}^d)$ is compactly embedded in $C(0,T;\mathbb{R}^d)$. The proof for the continuous embedding part is straightforward and is given in PDE ...
2
votes
1answer
88 views
Is the Sobolev Space $H^k(0,1)$ a banach algebra?
In Adams'book:Sobolev Spaces, I know that if $kp>n,\Omega\subset R^n$ is boundary domain and has cone property, then $W^{k,p}(\Omega)$ could see as a banach algebra. My question is that does it ...
1
vote
1answer
51 views
Is Duality map Odd?
Suppose $p\in (1,\infty)$ and $\frac{1}{p}+\frac{1}{p'}=1$. Let $J: W^{-1,p'}\rightarrow W_0^{1,p}$ be the duality map i.e. $$\langle f,J(f)\rangle=\|f\|_{W^{-1,p'}}^2\text{ and } ...
4
votes
1answer
142 views
Understanding the dual
There is an argument that disturbs me somewhat in Rabinowitz : Minimax methods in critical point theory. p.94
We are trying to prove that for certain functions, the Palais - Smale condition can be ...
2
votes
1answer
80 views
The Sobolev norm for vector-valued functions
For a compactly supported function $f: \mathbb{R}^n \to \mathbb{C}$, the Sobolev norm is defined by
$$\|f\|_s^2 = \int |\hat{f}(y)|^2(1+|y|^2)^sdy.$$
Here $\hat{f}$ is the Fourier transform of $f$, ...
4
votes
2answers
87 views
Compactness result PDEs
There is an argument that I see is used often in Evans PDE book, that I don't really get. We take a bounded sequence, say $(u_m) \in W^{1,q}(\Omega)$. By some functional analysis results, we know ...
2
votes
0answers
38 views
Inequality involving Bessel potential.
I'm not able to prove the following inequality: Fix $s>0$
$$\|fg\|_{H^s}\lesssim \|fJ^sg\|_{L^2}+\|gJ^sf\|_{L^2},$$
where $\widehat{J^sf}(\xi)=(1+|\xi|^2)^{s/2}\hat{f}(\xi)$ (Bessel potential).
...
3
votes
1answer
62 views
About a counterexample of an inequality？
I have known how to use the compactive argument to prove the inequality (1), i.e. $1\leqslant p<n$, $\Omega\subset R^n$ is a bounded domain,$\forall \varepsilon>0$, there is ...
1
vote
0answers
41 views
How to show that $u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n}))$?
If $u(t,x)\in L^{2}([0,T],H^{2}(\mathbb{R}^{n}))$, $\partial_{t}u \in L^{2}([0,T],L^{2}(\mathbb{R}^{n}))$, prove that
$$
u(t,x)\in C([0,T],H^{1}(\mathbb{R}^{n}))
$$
1
vote
1answer
109 views
Sobolev Embedding (Case: p=N)
Let $\Omega\subset\mathbb{R}^N$ be a regular bounded domain. Suppose $p=N$, then by Sobolev theorem, we have that for fixed $q\in [1,\infty)$ $$\|u\|_q\leq C\|u\|_{1,N}\ ,\forall\ u\in ...
1
vote
1answer
64 views
How to prove the density result？
How to prove that $C_c^\infty(R^n)$ is dense in $C^0(R^n)$,where the topology of $C^0(R^n)$ defined as follows
$u_k\rightarrow 0$ in $C^0(R^n)$ if and only if $\forall K\subset\subset R^n$,$sup_{x\in ...
4
votes
1answer
74 views
Poincaré's lemma with norm in $H_{0}^{1}$
I wonder why $\| u\|_{H_0^1} = \int_{\Omega} |Du|^2$ for u in $H_0^1(\Omega)$, with $\Omega = (-1,1)$? I might be wrong but isn't $\| u\|_{H^1} = \| u\|_{L^2} + \| Du\|_{L^2}$? How come that $\| ...
2
votes
1answer
48 views
question about sobolev inequality
I have two qeustions about Sobolev spaces:
Is there any Sobolev inequality that $Du$ bounded with $Lp$ norm $u$. For example
$$||Du||_{Lp}\leq||u||_{Lq}$$
and no in $W^{1,p}$. And my second question
...
2
votes
1answer
77 views
Singular Value Decomposition - what do I have to do?
Show that the Singular Value Decomposition of
$$
T\colon L^2([0,1])\to H^1([0,1]), x\mapsto\int\limits_0^t x(s)\, ds
$$
is given by
$$
\sigma_j=\frac{1}{(j-1/2)\pi}, v_j(x)=\sqrt{2}\cos((j-1/2)\pi ...
3
votes
2answers
125 views
Find adjoint operator of an operator T
I would like to find the adjoint operator of
$$
T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds.
$$
Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$.
I tried to find ...
2
votes
1answer
63 views
Existence and uniqueness of PDE with solutions in $W^{k,p}$ with $p \neq 2$?
I just realised that i have never seen the space $W^{k,p}$, $p\neq 2$, used in showing existence/uniqueness to some PDE. Usually books/lectures build up theory about $W^{k,p}$ (like certain compact ...
3
votes
1answer
45 views
Is $W_0^{1,p}$ weakly closed?
is $W_0^{1,p}(\Omega)$ weakly closed? $W_0^{1,p}(\Omega)$ is the closure of $C_0^{\infty}(\Omega)$ with respect to the norm of $W_0^{1,p}(\Omega)$, and I've been trying to figure out if it is true ...
2
votes
1answer
82 views
How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms?
How to prove that $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ and $\lVert u \rVert_{H^2}$ are equivalent norms on a bounded domain? I hear there is a way to do it by RRT but any other way ...
5
votes
1answer
169 views
Gradient of a Lipschitz function on a Riemannian manifold.
I'm referring to the article of D. Fischer-Colbrie and R. Schoen The structure of complete stable minimal surfaces in 3-Manifolds of non-negative scalar curvature (journal link, pdf).
In the proof ...
3
votes
1answer
65 views
Index of a Fredholm Operator on Paths
I'm a novice to analysis but I need to understand the following example. Any help would be greatly appreciated. This might be of interest to some because it gives a way of quantifying changes in ...
4
votes
1answer
110 views
Sobolev embedding for $W^{1,\infty}$？
From the Evans'PDE book I learned $W^{1,\infty}(U)$ coincide with Lipschitz continuous $C^{0,1}(U)$ with $U\in\Bbb{R}^n(n\geqslant1)$ is bounded and $\partial U\in C^1$. I wonder the counterpart ...
3
votes
1answer
96 views
Which domains make Sobolev inequality weaker?
Let $p<n$ and consider the Sobolev inequality on $W^{1,p}(\mathbb{R}^n)$ space:
$$\tag{1} \left\lVert u \right\rVert_{p^\star, \mathbb{R}^n} \le C\left\lVert \nabla u ...
-1
votes
1answer
60 views
When Dirac function is in $H^{-m}(R^n)$?
If Dirac function $\delta\in H^{-m}(R^n)$,please give the range of $m$?
2
votes
0answers
161 views
How to prove a version of Poincare inequality?
I want to use the contradiction argument and compact argument to prove the inequality below
$\forall\epsilon>0$,there exists $C_\epsilon>0$,$\forall u\in W^{1,p}(U)$,we have
...
1
vote
1answer
178 views
weak subsolution
Assume $u\in H^1(U)$ is a bounded weak solution of
$$-\sum_{i,j=1}^n(a^{ij}u_{x_i})_{x_j}=0 ~~~in~ U$$
Let $\phi:R\rightarrow R$ be convex and smooth,and set $w=\phi(u)$
Show $w$ is a weak ...
3
votes
1answer
89 views
Proof that the spectrum of the Dirichlet Laplacian is discrete
Let $\Omega\subset\mathbb{R}^n$ a open bounded set. The Dirichlet laplacian can be defined via it's closed semi-bounded form on $H^1_0(\Omega)$. The fact that it's spectrum is discrete is as far as I ...
2
votes
0answers
60 views
Bound for a Sobolev function in an integral
For a compact bounded set $\Omega$, for the expression
$$\int_\Omega \Delta u (\nabla u \cdot \nabla f)$$
where $u \in H^2$ and $f \in C^\infty$, is it possible to show that the expression is $\geq$ ...
2
votes
0answers
95 views
Sobolev space inequality
If $f\in H^2(\mathbb R^2)$, I want to show that
$||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$
How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm?
It ...
1
vote
1answer
336 views
Poincare Inequality
In page 290 of this book, Evans prove the Poincare inequality (Theorem 1) arguing by contradiction. Is there a direct proof of this theorem (Theorem 1) without arguing by contradiction?
2
votes
0answers
63 views
Convergence of Schwartz kernels implies convergence of operators
Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
4
votes
1answer
199 views
Rellich–Kondrachov theorem for traces
Let $W^{1,p}(\Omega)$ be the Sobolev space of weakly differentiable functions whose weak derivatives are $p$-integrable, where $\Omega \subset \mathbb R^n$ is a domain with Lipschitz boundary. Let ...
1
vote
0answers
20 views
Density of finite element functions in $W^{1,p}(\Omega)$
I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of ...
2
votes
1answer
40 views
Is $W^{k,p}$ a reflexiv Banach space?
Let $U\subset \mathbb{R}^n$ be an open bounded subset with smooth boundary. Is the Banach space $W^{k,p}(U)$ a reflexiv Banach space? If not, for what $k$ and $p$ is it reflexive?
2
votes
1answer
142 views
Sobolev inequality
If $f\in H^2(\mathbb R^2)$, I want to show that
$||f||_{L^\infty}\le c||f||_{H^2}$
$||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$
For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb ...
5
votes
1answer
137 views
Is there anyway to bound the $L^\infty$ norm by other $L^p$ norm?
If $f\in L^\infty(\mathbb R^2)$ (in my particular exercise, $f\in H^2(\mathbb R^2)$, the sobolev space), I want to bound $|f|_{L^\infty}= $ esssup $|f|\leq c|f|_{L^p}$ for some p, what kind of number ...
3
votes
1answer
142 views
Sobolev space exercise
I need to show $|f|_{L^\infty}\leq c|f|_{H^2} = c(\int_{\mathbb R^n} (1+|\xi|^2)^2|\hat f(\xi)|^2 d\xi )^{1/2}$, assume $f\in H^2(\mathbb R^2)$
I think I can trasnfer $f\ = \int \hat f(\xi)e^{2\pi i ...
1
vote
1answer
128 views
Poincaré inequality using $H^1$ seminorm
Does this inequality holds for Poincaré Inequality?
$$||v||_{L^2} \leqslant C_p |v|_{H^1}$$
and $$ |v|_{H^1} = ||v'||_{L^2} $$
where $| \dot~ |$ is the semi norm and $||\dot~||$ is the norm.
I'm ...
1
vote
1answer
70 views
Fourier transform of function defined on subset of $\mathbb{R}^n$
If I have a function $f:\Omega \to \mathbb{R}$ in $H^k(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is compact, then what is known about the Fourier transform $\hat{f}$? What space does it lie in? I ...
2
votes
1answer
75 views
Sobolev space $H^2$ norm in terms of gradient
Is it the case that
$$|v|_{H^2(\Omega)}^2 = |v|_{H^1(\Omega)}^2 + \int_\Omega\sum_{i=1}^n |\nabla v_{x_i}|^2?$$
I think yes but I have never seen anybody write it like this. I guess generalisations ...
3
votes
1answer
101 views
Question on proof in Evans PDE
This is on page 542 of Evans PDE book. The last inequality states that
$$\int_{U}{C(|Du|+1)|u|dx} \leq \frac{1}{2}\int_{U}|Du|^2dx + C\int_{U}{|u|^2+1 \ dx}$$
Where is this coming from? I think ...
1
vote
0answers
22 views
Unusual Compact Embeddings
Can anybody give a reference to the following two facts?
The embeddings $$H_0^{1,2}(\mathbb R^n)\to L^2(\partial B_1(0))$$ and $$H^{1,2}(\partial B_1(0))\to H^{1/2,2}(\partial B_1(0))$$ are compact?
...
1
vote
1answer
92 views
Distributional/weak time derivative basic question
Suppose we have $u \in L^2(0,T;H^1(\Omega))$, and $v \in L^2(0,T;H^{-1}(\Omega))$ is the weak time derivative of $u$, so by definition it satisfies
$$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$
...
1
vote
0answers
30 views
Minimum is attained in a subset of a Sobolev space
Let $\Omega \subset \mathbb R^n$. I have a functional of the form,
$$\int_{\Omega}f(x,u,\nabla u)dx$$
where $u \in W^{1,p}(\Omega, M)$, $M \subset \mathbb R^d$ is a compact, smooth Riemannian ...
0
votes
1answer
32 views
$W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$?
How can I prove that $W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$ if $s_1 > s_2 + n/4$ ? $W^{s,p}$ denotes a general Sobolev space for $s =0,1,2,\cdots$. The hook means a ...
0
votes
1answer
94 views
$u\in W^{1,p}(1,0)$ is equal s.e. to an absolutely continuous function?
I have a simple question on Sobolev space theory. Let $1\le p \le \infty. $How can one prove that $u\in W^{1,p}(1,0)$ is equal s.e. to an absolutely continuous function and that $u'$ exists a.e. and ...
1
vote
0answers
79 views
A continuous embedding.
If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
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 ...
