Questions tagged [sobolev-spaces]
For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.
3,342 questions
1
vote
0answers
27 views
$ L^p$ regularity for weak solutions of non homogeneous heat equation
Let $U$ be a compact $C^2-$manifold and suppose $v:U\times (0,T) \to \mathbb R$ is the weak solution of:
$\partial_t v= \Delta v+f$ where $f\in L^{\infty}(U\times (0,T))$
I 'm interested in the $L^...
0
votes
1answer
30 views
Embedding for homogeneous Sobolev spaces
Let $B_R$ be a ball of radius $R$ in $\mathbb{R}^d$. Under what conditions does the Sobolev embedding $\dot{W}^{1,p}_0(B_R) \hookrightarrow L^{p^*}(B_R)$ holds, where $p^* = \frac{np}{n-p}$?
Here $\...
2
votes
0answers
41 views
Norm inequality in Sobolev space $H^s$ as algebra for $s>n/2$.
Let $n\in \mathbb N^*$ and let $s>n/2$. Then the Sobolev space $H^s(\mathbb R^n)$ is a multiplicative algebra included in $L^\infty(\mathbb R^n)$ and we have
$
\Vert uv\Vert_{H^s(\mathbb R^n)}\le ...
0
votes
1answer
33 views
weak convergence of weak derivative in Sobolev Spaces
Short question cornerning some lectures notes in my current calculus of variation class:
Let $\Omega \subset \mathbb{R^n}$ be open and bounded. It is now stated that if $(\phi_j)_{j \in \mathbb{N}} \...
2
votes
0answers
44 views
eigenfunctions of laplacian on infinite strip
I am trying to solve
\begin{equation}
-\Delta{u} = \lambda \cdot u
\end{equation} with zero-boundary-conditions on the unbounded domain $\Omega = \mathbb{R} \times (0,\pi)$ for $\lambda \ge 0$ in the ...
1
vote
1answer
38 views
Understanding the minimal and maximal closed extensions of Laplace operator
Let's consider the following setting. $\Omega$ is a connected, riemmanian manifold with boundary (although i think there is no harm on thinking it is a bounded, close region of $\mathbb{R}^n$).
One ...
0
votes
1answer
17 views
Norm Inequality for 1 Dimensional Sobolev Space
Let $\Omega \subset \mathbb{R}$ be an unbounded domain and $u \in H_{0}^{1}(\Omega)$. By Sobolev Embedding Theorem for 1 dimensional space, we can obtain
$$S ||u||_{p}^{2} \leq ||u||_{H^{1}_{0}(\Omega)...
0
votes
0answers
17 views
Function in H(curl) $\cap$ H(div), but not in H1
it is well known, that for a non-convex domain $\Omega$ the space $H^1(\Omega, \mathbb{R}²)$ is a proper subset of $H(curl) \cap H(div)$.
Here, $H(curl) = \{v \in L²(\Omega)², \nabla\times v = \...
1
vote
1answer
54 views
Hardy-Littlewood Inequality for Sobolev spaces
After making the mistake of applying Hardy-Littlewood-Sobolev(H-L-S) for the infinity case, I was wondering if it is possible to bound it by a Sobolev norm.
Fix dimension to be $3$. H-L-S says that ...
2
votes
1answer
45 views
Does integral of function against all test function derivative imply function is equal to 0?
Howdy so I know that if
$\int fv = 0 $ for all test functions $v$ then $f=0$
If you have however
$\int fv_x = 0 $ for all test functions $v$ then does it mean that $f=0$?
I guess my thought is ...
2
votes
1answer
33 views
Weak convergence in $W^{1,p}(\Omega)$
I am a bit confused by weak convergence in Sobolev spaces. I am making this post to hopefully clarify some of my doubts.
Recall that in a Banach space, we say a sequence $x_n\in X$ converges weakly ...
2
votes
0answers
38 views
+50
Can we extend the Riesz potential convolution operator for the Laplacian to a continuous operator from $L^p$ to $\mathcal{S}'$ if $p\ge\frac{n}{2}$?
If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define:
$$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$
Then $K_n$ is locally ...
2
votes
1answer
44 views
Can we say anything about the first distributional derivatives of $g$, where $g$ is the solution to $-\Delta g =f\in L^p$ given by Riesz potential?
If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define:
$$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$
Then $K_n$ is locally ...
2
votes
0answers
30 views
$H^s(\mathbb R^d) \subset \bigcap_{2<p<\infty} L^p(\mathbb R^d)$ $0<s<1/2$?
Consider Sobolev spaces
$$ H^s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^2(\mathbb R^d) \}$$
where $\langle \cdot \rangle = (1+ |\...
2
votes
0answers
60 views
Sobolev Spaces subsets of each other?
I have recently started working with Sobolev Spaces and I wanted to ask the following:
Let $1 \leq p \leq \infty, \Omega \subset \mathbb{R}^d$. Does $ W^{n,p}(\Omega) \subset W^{1,p}(\Omega)$ hold?
...
1
vote
0answers
56 views
Compact Imbedding on weighted Sobolev Spaces
I am trying to solve the simple Eigenvalue problem
$-\Delta{u}=\lambda u, \; u = 0$ on $\partial{\Omega}$
in Sobolev spaces on a smooth exterior (and thus unbounded) domain $\Omega \subset \mathbb{...
2
votes
0answers
33 views
local approximation by smooth functions
Can someone explain to me where the yellow one comes from...
thank you
0
votes
0answers
15 views
eigenfunction representation with spline: show that coefficients fall faster than order of eigenvalues
I'm trying to understand the Proof of Theorem 3 in Bühlmann & Yu 2003 (Boosting with the $L_2$-Loss). The paper considers some projection matrix $S$ corresponding to a smoothing spline of degree $...
-1
votes
1answer
25 views
Question on norm on Sobolev Space
If $u\in W^{k,p}(\Omega)$ we define its norm as
$$
\Vert u \Vert_{W^{k,p}(\Omega)}=
\begin{cases}
\displaystyle
\left(\sum_{|\alpha|\le k}\displaystyle\int\limits_\Omega |D^\alpha u|^p\mathrm{d}x\...
2
votes
1answer
39 views
Uniqueness of weak derivative
In this result, I understand almost everything but I don't understand why we using
$\Omega' \Subset \Omega$ what is the major role of this why not we directly use $\Omega$.
thank you
2
votes
1answer
42 views
Upper estimate for an integral in norms
Is there any reference to obtain an upper bound for
$$
\int_{\Omega}v_{t}^2(v^2+1/v^2)dx
$$
where $v\in H^{2}(\Omega)\cap H_{0}^1(\Omega) \setminus \{0\}$, $v_{t}\in H_{0}^1(\Omega) \setminus \{0\}$ ...
0
votes
1answer
42 views
Density of $C_c^\infty([0,1])$ in $H_0^1(0,1)$
I'm trying to prove that $C_c^\infty([0,1])$ is dense in $H_0^1(0,1) = \{ f \in H^1(0,1) : f(0) = f(1) =0\}$ for the usual Sobolev norm $\Vert f \Vert = \Vert f \Vert_{L^2(0,1))} + \Vert f' \Vert_{L^2(...
0
votes
1answer
27 views
Exercise about the trace of sobolev functions
Let $Q_1$ and $Q_2$ be two open squares in $\mathbb{R}^2$ whose closures have an edge - say L - in common. Let be $u_i\in W^{1,p}(Q_i)$ for $i=1,2$ and for such $p\in[1,+\infty]$.
Suppose that $Tr(...
1
vote
1answer
34 views
maximum of trace operator and 0
Let $ n \in \mathbb{N}, \ \Omega \subset \mathbb{R}^n $ be a bounded domain with Lipschitz-boundary and $ S:H^1(\Omega) \to L^2(\partial \Omega) $ the trace Operator.
For $ u \in H^1(\Omega): \quad$
...
0
votes
0answers
20 views
Norm convergence of Fourier series in the dual of a Sobolev space
If $\mathbb{T}$ is the 1-torus and $1<p<\infty$, then for every $f$ in the Sobolev space $W^{1,p}(\mathbb{T})$ we have that the Fourier series of $f$ converges in the $W^{1,p}(\mathbb{T})$ to $f$...
2
votes
0answers
20 views
Does the Sobolev space $W^{1,p}(\Omega), p>2$ has a monotone basis?
A Shauder basis in a Banach space is monotone if $\|P_{n}f\|\leq\|f\|,$ where $P_{n}$ is the projection to the sum of the first n elements of the basis. For Hilbert spaces this is always the case if ...
1
vote
1answer
29 views
Product from function in $W^{1,2}_0(\Omega)$ and function in $W^{1,2}(\Omega)$
I was wandering what i say about product
$$
u\eta
$$
where $u\in W^{1,2}(\Omega)$ and $\eta\in W^{1,2}_0(\Omega)$.
In particular, when i can say that
$$
u\eta\in W^{1,2}_0(\Omega).
$$
Is it necessary ...
2
votes
1answer
35 views
Weak derivative of a Sobolev function in unbounded domains
I have following setup:
$f\in W^{1,1}(\mathbb{R}^3)$ with $f\geq 0$ and $\int_{\mathbb{R}^3}f=1$, how can I show that the weak derivative of $\tilde{f}(x):=\int_{\mathbb{R}^2}f(x,y,z)\,\text{d}y\text{...
2
votes
1answer
24 views
Exercise on Sobolev Spaces: Show that this function belong to $W^{1,\infty}$
Let $u \in C^0(I)$ be a bounded function on the open interval $I=(a,b)
\subset \mathbb R$. Suppose that there exists a partition $a=t_0 \lt
t_1 \lt \dots \lt t_n=b$ such that:
$f \in C^1((...
5
votes
1answer
95 views
+50
Sobolev embedding for the $L^q$ norm.
Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.
2
votes
1answer
26 views
Sobolev integrability implies weak flatness?
Let $B_1\subset \mathbb R^n$ be the unit ball, and suppose $u\in W^{1,2}(B_1)$, i.e. $\int_{B_1} u^2+|Du|^2\,dx\le K<\infty$. Moreover, suppose $u$ is nonnegative, with $ess\inf u=0$.
Let us also ...
0
votes
0answers
18 views
A parabolic Morrey-Sobolev inequality
Let $B \subset \mathbb R^2$ be the unit ball and $T>0.$ Let $u \in W^{2,1}_p(B \times [0,T]),$ that is $u \in L^p(B \times [0,T])$ and we also have,
$$ \partial_t u, \nabla u, \nabla^2 u \in L^p(B \...
0
votes
0answers
23 views
Gauss - Green Theorem on Sobolev Spaces
Can some one explain please what is exactly of this theorem i dont understand even one word
i don't understand the notations also please ..
thank you
1
vote
0answers
38 views
Where I get video lectures of Sobolev spaces
i am start studying Sobolev Spaces By the book name Partial Differential Equations by Laerence C. Evans...
but while i am studying i have so many doubts and i asked one of my professor he said that ...
2
votes
1answer
35 views
Functional Space Inequality for Sobolev Space and Lp Space
Let $X = C_{0}(\Omega) := \{ u \in C(\overline{\Omega})\,|\,u|_{\partial\Omega}=0\}$ and define $F : X \to X$ as Lipschitz continuous function and $F(0) = 0$. Let $\Omega\subset \mathbb{R}^{N}$ be a ...
0
votes
2answers
70 views
Integration by parts in $\mathbb{R}^n$
Let $U \subset \mathbb{R}^n$ be open. Let $f \colon U \to \mathbb{R}$ be a function of class $C^1$, and let $\phi\colon U \to \mathbb{R}$ be a function in $C_c^\infty (U)$. Is it true that $$\int\...
0
votes
1answer
36 views
Weak convergence of sequence in Sobolev space implies uniform convergence
Does weak convergence in a Sobolev space implies uniform convergence?
In the above question, a proof by contradiction is given from which it follows that if a sequence in $W^{1,p}$ over a nice, open ...
0
votes
0answers
21 views
Closed operator in sobolev spaces
So suppose we look at the operator $S:H_0^1(0,1) \rightarrow L^2(0,1), \ u\mapsto u' $,
where $H_0^1(0,1)$ denotes the closure of the infinitely differentiable functions compactly supported in $(0,1)...
1
vote
0answers
35 views
On boundedness in fractional Sobolev norm
Let us asssume that a sequence $(v_n)$ satisfies
(i) $v_n\in{\rm W}^{2,\infty}(0,1)$,
(ii) $||v_n||_{{\rm H}^{\theta}(0,1)}$ is bounded for some $0<\theta<1$.
Is it true that then the ...
1
vote
1answer
61 views
Norm of dual space of $H_0^1$
Let $H^{-1}$ denote the dual space of $H_0^1(\Omega)$. Then every $f \in H^{-1}$ can be represented as
$$f(u) = \int_\Omega f^0u\ dx + \sum_{k=1}^n \int_\Omega f^k \partial_k u\ dx$$ for some ...
0
votes
1answer
40 views
Question about to Weak derivative of $|x|$
As I know that the function $f(x)=|x|$ is not differentiable.but in the weak sense it has weak derivative
my question is it again weak derivative exists for this function
I.e.,
suppose $f_1$ is ...
3
votes
1answer
66 views
What is the motivation of defining weak derivative as it is?
I've been reading lately about reproducing kernel Hilbert spaces (RKHS) and Gaussian processes (GP) and during my studies I came across with the concept of weak derivative and Sobolev spaces. I have ...
0
votes
1answer
19 views
Supnorm Defined on set of all uniformly continuous functions is Banach space
If we define Supnorm on $C(\bar{U})$ is
$$||f||_{C(\bar{U})}:=\sup_{x\in U} |f(x)|.$$ on $C(\bar{U})$ is Banach space
what I know is :
I'm proving this normed space
since $||f||_{C(\bar{U})}=\...
0
votes
1answer
23 views
Supnorm on uniformly continuous functions
While I am studying Evans book He define Supnorm on $C(\bar{U})$ is
$$||f||_{C(\bar{U})}:=\sup_{x\in U} |f(x)|.$$
where $f:U\to \mathbb{R}$ is bounded and continuous
and $C(\bar{U})=\{f\in C(U) $ ...
0
votes
0answers
29 views
Integral Equation and Fredholm Alternative
I am learning about functional analysis at the moment and I have difficulties grasping the connection to integral equations or differential equations. For simplicity, let us consider the following ...
1
vote
1answer
63 views
Prove that space is Hilbert
Let $$H_0^1(0,1)=\{f\in W^{1,2}(0,1):f(0)=0\}$$
and a norm $$\| f\|=\left (\int_0^1 |f'(x)|^2\mathrm{d}x\right )^{1/2}$$
be given. I want to show that if a sequence $(u_n)_{n\in\mathbb{N}}$ in $(H_0^...
0
votes
0answers
19 views
Approximation of trace zero Sobolev functions
Let $U\subset\mathbb{R}^n$ open and bounded with $\mathcal{C}^\infty$ boundary. I want to prove that for any $f\in H_0^1(U)\cap H^k(U)$, there is a sequence $u_n\in\mathcal{C}^\infty(\overline{U})$ ...
5
votes
2answers
44 views
Convergence in $C([0,T_0],L^2)$ and uniform boundedness in $C([0,T_0],H^2)$ gives convergence in $C([0,T_0],H^1)$.
Let $\Omega$ be a compact set of $\mathbb{R}$ and $s\geq 1$. Let
$$
v_n\in C([0,T_0];H^{s+1}(\Omega)).
$$
Also $\sup_{t\in[0,T_0]} ||v_n||_{H^{s+1}(\Omega)}\leq M$, $M$ is a constant.
We are also ...
1
vote
1answer
110 views
proving an interpolation inequality in $L^p$ norms
Let $1\le p \le \infty$. Prove that for all $\epsilon >0$ there exists a constant $C>0$ such that
$$\|u'\|_{L^p(\mathbb{R})}\le \epsilon \|u''\|_{L^p(\mathbb{R})}+C\|u\|_{L^p(\mathbb{R})} \;\; \...
0
votes
0answers
20 views
Prove a given function is an extender form $W^{1,p}(\Omega)$ to $W^{1,p}(\mathbb{R}^2)$
I am trying to prove that given the set $\Omega=\{(x,y) \in \mathbb{R}^2 | y > \sin(x) \}$, the function $E: W^{1,p}(\Omega) \to W^{1,p}(\mathbb{R}^2)$ with $Eu(x,y)=u(x,|y-\sin(x)|+\sin(x))$ is an ...
