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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.

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$ 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^...
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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 $\...
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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 ...
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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}} \...
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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 ...
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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 ...
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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)...
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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 = \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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+ |\...
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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? ...
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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{...
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33 views

local approximation by smooth functions

Can someone explain to me where the yellow one comes from... thank you
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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 $...
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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\...
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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
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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 $v\in H^{2}(\Omega)\cap H_{0}^1(\Omega) \setminus \{0\}$, $v_{t}\in H_{0}^1(\Omega) \setminus \{0\}$ ...
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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(...
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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(...
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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$ ...
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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 $f$...
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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 ...
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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 ...
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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{...
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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((...
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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.
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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 ...
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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 \...
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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
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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 ...
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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 ...
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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\...
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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 ...
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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)...
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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 ...
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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 ...
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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 ...
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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 ...
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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})}=\...
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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) $ ...
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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 ...
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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^...
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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
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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
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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})} \;\; \...
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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 ...