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.

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Equivalent definitions of Poincare inequality

I can't seem to find this anywhere but I can find that there are (at least) two definitions of the Poincare inequality. One is $$ \int_\Omega |f|^2 dx \leq c\int_\Omega |\nabla f(x)|^2 dx $$ for $f \...
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Brezis book, Functional analysis, Sobolev spaces and PDE, problem 8.30

Let $k \in \mathbb{R}$, $k \neq 1$, consider the space $$V= \{ v \in H^1(0,1): v(0) = k v(1)\}$$ and the bilinear form $$B(u,v) = \int_0^1 \left(u'v' + uv\right) ~dx - \left(\int_0^1 u\right) \left(\...
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Apply Arzela-Ascoli theorem to unifomly bounded sequence in $H^1$

Let $\{u_{i}\}$ be sequence of smooth function defined on $\Bbb{R}^n$ such that $\|u_i\|_{L^2(\Bbb{R}^n)}$ is uniformly bounded in $i$ and $\|\nabla u_{i} \|_{L^{^2}(\Bbb{R}^n)}$ is also uniformly ...
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How to the estimate as application of Strichartz Estimates?

RecallA pair $(q,r)$ is admissible if $q\geq 2, r\geq 2$ and $\frac{2}{q} = N \left( \frac{1}{2} - \frac{1}{r} \right),(q,r,N)\neq(\infty,2,2).$ Strichartz estimates Let $\phi \in L^2(\mathbb R^N),$...
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1 vote
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Elliptic regularity with right-hand side in $H^{-1/2}$

If $\mathbf{f} \in H^{1/2}(\Omega)$ for a bounded domain $\Omega$ with smooth boundary, does the elliptic regularity for the Laplacian guarantee that the solution to the elliptic problem in divergence ...
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Weak convergence in $H^1(\mathbb{R}^N)$ implies weak convergente in $H^1(B)$

Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite ...
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1 vote
1 answer
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Proof of the embedding of time dependent Sobolev spaces

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
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  • 567
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1 answer
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Hölder continuity on Sobolev space $H^1(a,b)$

Let $H^{1}(a, b)=W^{1,2}(a,b)$ be the Sobolev space, that is, the functions $f\in L^2$ such that $f' \in L^2$, where $f'$ denotes the weak derivative of $f$. This is equivalent to $$f(x)=c+\int_a^x f'...
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Proving $||f||^2_{L^2(\mathbb R^2)}\le 10||f||_{L^1(\mathbb R^2)}||\nabla f||_{L^2(\mathbb R^2)}$ with $f\in C^1,L^1\cap L^2$ and $\nabla f\in L^2$.

Prove that there exists a universal constant $K<10$, for all $C^1$ function $f : \mathbb R^2 \rightarrow\mathbb R$, if $f \in L^1 (\mathbb R^2)\cap L^2(\mathbb R^2)$ and $|\nabla f| \in L^2(\mathbb ...
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Prove the uniqueness of $u\in H_0^1(\Omega)$ with $\Delta u=\vert u\vert^{q-1}u+f$ in $\Omega$ with $\Omega$ as a bounded domain with smooth boundary.

Problem: Let $\Omega\subset\mathbb R^2$ be a bounded domain with smooth boundary. Prove that, for all $p>1$ and $1\le q<\infty$, for all $f \in L^p(\Omega)$, there exists a unique $u\in H_0^1(\...
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Showing $\lVert \Delta u \rVert_{L^2} + \lVert u \rVert_{L^2}$ is equivalent to $\|u\|_{H^2}$ norm for $H^2$ space

This question has been asked here showing $\lVert \Delta u \rVert_{L^2(U)} + \lVert u \rVert_{L^2(U)}$ is equivalent to $\|u\|_{H^2(U)}$ norm for $H^2(U)$ space assuming the domain $U$ is sufficient ...
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Elliptic regularity for Poisson equation

Let $U \subset \Bbb{R}^n$ be an open bounded smooth domain, let $u \in H^1_0(U)$ satisfy the Poisson equation: $$-\Delta u = f \tag{*}$$ with $f \in L^{2}(U)$, by the classical elliptic regularity ...
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Some questions in the compactness of Sobolev space and Holder space

Let $B_1$ be the unit ball in $\mathbb{R}^2$. Define $A=\{u \in W^{1,2}(B_1) : \|u\|_{ W^{1,2} } \leq 1\}$, $B= \{u \in L^{2}(B_1) : \|u\|_{ L^2 } \leq 1\}$, $C=\{u \in C^{0,\alpha}(B_1) : \|u\|_{ ...
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  • 523
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1 answer
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Sobolev inequality for cubes

Consider the fallowing result from Evans, 2010, page 279: Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ is $C^1$. Assume $1 \leq p < n$, and $u \in W^{1,p}(U)$. Then ...
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2 answers
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Is it possible to find the maximum of $r\ln\left(\ln\left(1+\frac{1}{r}\right)\right)$ for $r\in(0,1)$?

I was working on Chapter 5, Problem 14 of Evan's Partial Differential Equations 2nd ed and to prove integrability of $\ln \left (\ln \left (1+\dfrac{1}{|x|}\right )\right )$ on the unit $n$-ball, ...
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What is the relationship between weak derivatives and the decay of the coefficients in the eigenfunction expansion?

Suppose we have a function $f: [0,1] \to \mathbb{R}$ that has $2\beta$, $\beta \in \mathbb{N}$, weak derivatives. Defining $e_k(x) := \sqrt{2}\sin(k \pi x)$ the orthonormal sine-basis of $L^2([0,1])$, ...
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2 votes
1 answer
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Relationship between Sobolev-Slobodeckij spaces and Besov spaces

I am trying to understand these two different ways of defining fractional Sobolev spaces. In particular, I want to determine embeddings or equality between the Besov spaces $B^{s}_{p,p}$ and the ...
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-3 votes
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Weak derivative of power function

Let $f\in {W}^{1,p}(0,1)$ ($1\leq p\leq\infty$) such that $f>0$ a.e., and let $g = f^\alpha$ ($0<\alpha<1$). I am wondering if $$g'=\alpha f^{\alpha-1} f'\ (weak)\qquad a.e. ?$$ Can somebody ...
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relation between the norm of Sobolev space $H_1$ and $L^p$ norm for non-increasing radial function

I am interested to find $$\sup\|u\|_{p}^{p},$$ where the sup is over non-increasing radial functions $u$ on the unit ball $B_{1}$ of $\mathbb{R}^{n}$ such that $$\|u\|_{H_1}^2 <r$$ for some $r >...
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How to bound non-linear terms of integral operator in Sobolev Space

Let $I=[0,1]$ and consider the function $F:H^1(I)\to \mathbb R$ given by $$F(u)=\int_I \bigr( u(t) \bigr)^4 dt. $$ This makes sense, because $H^1(I)\hookrightarrow C^0(I).$ I want to understand the ...
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Function in $H^1(\mathbb{R})$

Consider the function $$f=|x|^pe^{-x^2},$$ where $p$ is a real constant. The function $f$ is in $L^2(\mathbb{R})$ iff $p>-1/2$. The function $f$ is in $H^1(\mathbb{R})$ iff $p>1/2$ or $p=0$. I ...
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1 answer
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Bessel inequality for $H^1$ function

Let $U\subset \Bbb{R}^n$ be the bounded sufficient smooth open set, let $g\in H^1(U)$, and let $\{w_k\}$ be a sequence of the orthonormal basis for $L^2(U)$ and orthogonal basis in $H^1(U)$ (which can ...
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1 vote
1 answer
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Lax-Milgran theorem with the Poisson equation

Let $ \Omega $ be a bounded domain in $\mathbb R^3$ with smooth boundary. Consider the Poisson equation $$ -\Delta u=f $$ where $ f\in C_0^{\infty}(\Omega) $ and $f$ is null outside $\Omega$. I'm not ...
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1 answer
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Sobolev embedding implies lowerbound on the volumn of the ball

Assume Sobolev embedding: $$H_{1}^{1}(M) \subset L^{n /(n-1)}(M)$$ holds for the compact Riemannian manifold $(M^n,g)$, then we can get $$\operatorname{Vol}_{g}\left(B_{x}(r)\right)^{(n-1) / n} \leq C ...
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1 vote
0 answers
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Mean value theorem in Sobolev spaces

Let $\Omega$ be open and sufficiently smooth, $0\leq \tau \leq T$ and consider $u \in L^2(0,T; H^1(\Omega)), \partial_t u \in L^2(0,T;L^2(\Omega))$. Can we conclude $$\int_0^\tau||u(\tau)-u(t)||^2_{L^...
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0 answers
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Gradient of a functional defined on an Hilbert space (with respect to a $W^{1,2}$ inner product)

$\newcommand{\R}{\mathbb R}$ Consider the Hilbert space $X = W^{1,2}(\R)\oplus W^{1,2}(\R)$ (Sobolev spaces). I define a function $F:X\to \R$ as $$F(u,g)= \int_\R u(t)\partial_tg(t) dt.$$ $F$ is ...
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1 vote
0 answers
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Equivalent definitions of Sobolev space on manifold and references

It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$: D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm. D2. All functions ...
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  • 1,073
1 vote
1 answer
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How to deduce from these inequalities that $x_n\to 0$ in $W_0^{1, p}(\Omega)$?

Let $p, h, d\in\mathbb{R}, h>0, p>1$ and $n\in\mathbb{N}, n\ge 2$. Let $(x_n)_n\subset (W_0^{1, p}(\Omega),\|\cdot\|)$. During the class of today, the lecturer said that the inequalities $$\|x_n\...
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1 answer
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Exercise 8.19 - lower semicontinuous - l.s.c - Brezis

I'm sorry, but I don't know how to do the second part of item 1 of this exercise: Show that $\phi$ is l.s.c, that is, show that for each $\lambda \in \mathbb{R}$ set $$ [\phi \leq \lambda ] = \lbrace ...
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Examples of open and bounded sets in $\mathbb{R}^d$ satisfying the uniform cone condition

Cone conditions on domains are extensively used in the proof of the Sobolev embedding theorem. Somewhat less common in that respect is the uniform cone condition, which Adams and Fournier (Sobolev ...
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2 votes
0 answers
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How to prove that normed space is complete?

$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
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  • 521
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minimal induced norm on sobolev space

I am new to functional analysis and induced norms. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a challenging question for me. Could ...
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2 votes
1 answer
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A coercive bilinear form

Let $\alpha > 0$ and $X = [H^{2}(\Omega)\cap H^{1}_0(\Omega)] \times H^{1}_{0}(\Omega)$. Find $\lambda_0 > 0$ for which the bilinear form $B: X \times X \rightarrow \mathbb{R}$ given by $$ B((u,...
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Power of Sobolev function: if $ u \in H^2(\Omega) $, will $ |u|^{1+p} \in H^s(\Omega) $ for some $ s>1 $? Assume that $ p \in (0, 1) $.

Here, $ \Omega \subset \mathbb{R}^n $ is a bounded domain which can be as nice as you want (such as $ \mathbb{T}^n $ or $ B_r(0) $). I have an $ H^2 $ funciton $ u $ and I want to know if there is ...
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Can we use Newton-Leibnitz for $W^{1,p}$ function

For a function $u\in W^{1,p}$, we always use a sequence of $C^1$ function to approach it and derive the consequences. However, can we just claim for a.e. x, y: $$u(x)-u(y)=\int_0^1 Du(y+t(x-y))\cdot (...
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1 vote
0 answers
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Interpolation of Bochner-Sobolev spaces

Does anyone know how to prove or has an explicit reference for interpolation inequalities of the form $$ \|f\|_{H^{l}(0,T;H^{(1-l)}(\Omega))} \leq C \|f\|_{H^{1}(0,T;L^2(\Omega))}^l\|f\|_{L^2(0,T;H^1(\...
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1 vote
0 answers
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Composition $F\circ g$ of Sobolev function $g \in W^{k,2}(\mathbb R^n)$ with smooth map $F$ is Sobolev

$\newcommand{\R}{\mathbb R}$ Let $F: \R\to \mathbb R$ be a smooth function such that $F(0) = 0$. Now consider $g\in W^{k,2}(\R^n)$ (Sobolev space) with $k>n/2$. Is it true that $F\circ g\in W^{k,...
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5 votes
1 answer
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Remark 11 - Theorem 9.12 (Morrey) - Brezis - Pag 282 [closed]

I'm not sure about Remark 11. (i) Why is $\mathbb{R}^{N}\setminus A$ dense in $\mathbb{R}^{N}$??? (ii) How to ensure that the ongoing representative he defined is unique?
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Sobolev embedding for stochastic processes

Let's say I have some stochastic process $u_t \in L^2 ((0,T), H^2 (\Omega))$ with $\Omega$ as regular as you want in $\mathbb{R}^2$. If $E\left[ \int_0 ^ t ||u_t||_{H^2} ^2 \right] \leq C$, do I ...
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  • 2,753
1 vote
1 answer
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Reference for Sobolev-Hölder embedding for unbounded domains

I would like to know whether the following is true and references: If $\Omega\subset\mathbb R^n$ is open (not necessarily bounded) and $k = n/p + r+\alpha$, and $\alpha \in (0,1), r\in \mathbb N, k\...
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0 answers
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About a remark in Evans PDE, in the regularity of elliptic equation

On Page 341, after Theorem 5 (Higher boundary regularity) in Chapter 6.3 (Regularity), Evans states that Remark. If $u$ is the unique solution of $$ \left\{\begin{aligned} L u=f & \text { in } U \...
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4 votes
0 answers
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Derivative of a map $f:\mathbb R \to \ell^2 $ to a separable Hilbert space vs derivative of each component of the Hilbert basis.

${\newcommand{\R}{\mathbb{R}}}$ Let $\ell^2 $ be the Hilbert space of square summable sequences of real numbers. Consider a map $f: \R \to \ell^2$, that has components $f_n:\R\to \R$, i.e. $f(t)=\{...
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0 votes
1 answer
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About the continuity of the integral on the boundary of a ball of a $H^1$ function

I’m considering a $H^1$ function u on a open domain D. Is the integral: $$ \int_{\partial B_r(x)} u \hspace{2pt}dH^{n-1}$$ continuous with respect to x? I tried to prove that it’s differential by ...
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2 votes
0 answers
62 views

Assumptions in Schauder Fixed Point Theorem

I have a - maybe slightly stupid - question about the Schauder-Fixed-Point Theorem. The formulation I have in mind is: Let $A$ be a closed, convex, nonempty subset in a Banach space $(X,\|\cdot\|)$, ...
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  • 158
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1 answer
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Density of $[H^2(\Omega) \cap H^1_0(\Omega)] \times H^1_0(\Omega)$ in $H^1_0(\Omega) \times L^2(\Omega)$

I am reading Evans' PDE proof of theorem 6, section 7.4.3, page 444 where the following is said to be "clear" $[H^2(\Omega) \cap H^1_0(\Omega)] \times H^1_0(\Omega)$ is dense in $H^1_0(\...
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  • 479
0 votes
1 answer
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Prove a certain functional is unbounded from above

Let $\lambda\in\mathbb{R}$, $2<p\leq 2^*=(2N)/(N-2)$. For $u \in H^{1}_0({\Omega})$ where $\Omega$ is a domain of $R^N$. Define: $$ \varphi(u) = \displaystyle\int_{\Omega} \frac{1}{2}|\nabla u|^2 +...
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  • 179
0 votes
2 answers
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The relations between $ L^2 $ and $ H^{-1} $.

Let $ \Omega $ be a bounded domain in $ \mathbb{R}^d $. Suppose that $ f\in L^2(\Omega) $. Define $ \phi\in H^{-1}(\Omega) $ such that \begin{align} \langle\phi,u\rangle_{H^{-1}\times H_0^1}=\int_{\...
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0 answers
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If $\|(x_n, y_n)\|_W\to +\infty\quad\mbox{ as } n\to +\infty,$ we can assume WLOG that $\|x_n\|_{W^{1, p}}\to +\infty$?

Let $\Omega$ be an open bounded domain and let $p, q>1$. Consider $(x_n)_n\subset W^{1, p}(\Omega)$ and $(y_n)_n\subset W^{1, q}(\Omega)$. Defined $$W= W^{1, p}(\Omega)\times W^{1, q}(\Omega)\quad\...
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  • 807
0 votes
0 answers
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Reproducing kernel hilbert spaces: variation norm in arbitrary dimensions

I am currently dealing with the topic of reproducing kernel Hilbert spaces (RKHS) given the draft book of Francis Bach. As a background knowledge for my current problem define: \begin{align} &H_1=\...
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  • 53
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0 answers
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If $(u, v)\in W^{1, p}(\Omega)\times W^{1, q}(\Omega)$, what does the notation $|(u, v)|$ mean?

Let $\Omega$ be an open bounded domain, $p, q>1$ and let $W= W^{1, p}(\Omega)\times W^{1, q}(\Omega)$. Let $(u, v)\in W$. What does the notation $|(u, v)|$ mean? It is in my math class notes, but. ...
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