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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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closed subspace of $W^{1,2}(0,1)$

let $V_{C} = \{u \in W^{1,2}(0,1)\,\, | \,\, u(1) = Cu(0)\}, \,\, C \in \mathbb{R}$ show that $V_C$ is a closed subspace of $W^{1,2}(0,1)$ I got stuck trying to solve this here's my work : if we let ...
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11 views

Notation of source function for weak formulation of Poisson's equation

For $f \in L^2$ weak formulation of Poissons equation reads: Find $u \in H_0^1$ such that $$\int \nabla u \cdot \nabla \varphi \, \mathrm dx= \int f \varphi \, \mathrm dx \quad \forall\varphi \in ...
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16 views

Boundary value of continuous representative of a Sobolev function

Let $f \in H^1(a,b)$, where $a<b$. Then there exists a continuous representative $\tilde{f}(x) = \int_a^x f'(x) \ dx + c $ where $f'\in L^2(a,b)$ is the weak derivative of $f$ and c is a constant. ...
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1answer
38 views

On DiPerna-Lions compactness' arguments.

In DiPerna Lions, Ordinary differential equations, transport theory and Sobolev spaces (1989), the authors used topological arguments that remains obscure to me. Page 515 the authors used an ...
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29 views

Multiplication operator from Sobolev space to $L^p$ space is compact?

Let $M_q: W^{1,\tilde{p}}(\mathbb{R}^2) \rightarrow L^p(\mathbb{R}^2)$, where $\tilde{p}$ is the Sobolev conjugate of p, $\left(\frac{1}{\tilde{p}}=\frac{1}{p}-\frac{1}{2}\right)$. What we know is ...
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1answer
9 views

Dual of Sobolev space $H^{s}(\mathbb{R}^n)$ Taylor Michael.

Why the dual of $H^{s}(\mathbb{R}^n)$ is $H^{-s}(\mathbb{R})$? I know that dual of $H^{s}(\mathbb{R}^n)$ is $\left\{T:H^{s}(\mathbb{R}^n)\to \mathbb{C}:T \text{ bounded and linear functional} \right\}$...
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1answer
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Showing that $u\in H^k(\mathbb{R}^n)\Leftrightarrow <\xi>^k\hat{u}\in L^2(\mathbb{R}^n)$ Taylor Michael.

Note: $<\xi>=(1+|\xi|^2)^{1/2}$ Why $u\in H^k(\mathbb{R}^n)\Leftrightarrow <\xi>^k\hat{u}\in L^2(\mathbb{R}^n)$? To think that to use the density of the space of Schwartz in the space of ...
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approximating functions in $W^{1,2}$ with Lipschitz functions with image contained in fixed ball

Fix $r>0$. For $h>0$ let $u_h\in W^{1,2}(B_h,B_r)$, where $B_h$ and $B_r$ are the ball of radius $h$ and $r$ centered at the origin in $\mathbb R^n$, respectively. I want to approximate the $...
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1answer
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Properties of Functions in $W^{1,p}(\Omega)$ and Their Weak Derivatives

Let $\Omega \subset \mathbb{R}^{N}$ and $W^{1,p}(\Omega)$ be Sobolev Space. Then we let $u\in W^{1,p}(\Omega)$ and define (i) $u^{+} := \max\{0,u\}$ (ii) $u^{-} := \max\{0,-u\}$ Then, we claim that $u^...
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1answer
35 views

Definition de $L^1((0,T);W^{1,\infty}(\mathbb R))$.

I have some trouble about the definition of the space $L^1((0,T);W^{1,\infty}(\mathbb R)$, on one side the formal definition given by Bochner consists in the set of measurable functions $u$ from $(0,T)...
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29 views

Composition of Sobolev map and smooth function.

Let $\Omega = S^1 \times [a, b] $ and $f\in C^\infty (\Omega )$. When is $\;H^s(\Omega, \Omega) \rightarrow H^s(\Omega, \Omega), u \mapsto f \circ u$ well defined and smooth? I've only found results ...
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1answer
45 views

Why chose $H^{-1}(\Omega)$ instead of $L^2(\Omega)$?

For given $f \in L^2(\Omega)$ Poisson's equation reads $$- \Delta u=f \quad \text{on }\Omega.$$ So the variational problem becomes: For given $f \in H^{-1}(\Omega)$ find $u \in H_0^1(\Omega)$ such ...
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1answer
17 views

Strict inequality of the H1 seminorm of a curve emerging from a convex set and its convex projection

Let $\gamma(t) \in W^{1,2}([0,1]; \mathbb{R}^d)$ such that $\gamma(0) \in C$, with $C \subset \mathbb{R}^d$ some closed convex set, and $ \gamma(t) \not \in C, \ \forall t\in (0,1]$. Furthermore we ...
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PDE's as minimisation problems

I am reading through Chapter 8 of Brezis. Particularly Section 8.4, which deals with some examples of boundary value problems on an interval. For the most part I understand the steps in the proofs ...
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22 views

weak solution to non-homogeneous initial-boundary value problem

Let $\Omega\subset \mathbb{R}^n$ be a bounded Lipschitz domain, and consider a general linear parabolic equation of the divergence form: $$ u_t-\partial_i(a^{ij}\partial_ju)+b^i\partial_i u+cu=f, \...
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2answers
47 views

If compact embedding $H^1_0(\Omega) \hookrightarrow H^{-1}(\Omega)$ is not surjective, how can $H^{-1}(\Omega)$ be the dual of $H^1_0(\Omega)$?

Let $\Omega$ be bounded with Lipschitz boundary. By Rellich compactness, $\iota: H^1_0(\Omega) \hookrightarrow L^2(\Omega)$ is compact embedding. It is also dense. By Riesz representation, $L^2(\...
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1answer
72 views

Extension of a Sobolev Function

Let $\Omega _1$ and $\Omega_2$ are smooth open sets and $A= \partial \Omega_1 \cap \partial \Omega_2.$ Let $\Omega ^\prime = \Omega_1 \cup \Omega_2 \cup A$ be an open set. Let $f$ is defined on $\...
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Does anyone know any books provide complete picture of mathematical spaces, hierarchical view, relations, natural progression among them

I was wondering if anyone is aware of any books that provide a comprehensible and complete picture of mathematical spaces? I've looked into many real analysis as well as complex for a complete list of ...
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Is the definition of the Sobolev space H^1(M) on a compact manifold that simple?

In https://hebey.u-cergy.fr/NotesSharpSP.pdf right at the beginning Hebey says Given $(M,g)$ a smooth compact $n$-dimensional Riemannian manifold, one easily defines the Sobolev spaces $H^p_k(M)$,...
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Looking for a convergent sequence in $W^{1,2}[0,\infty)$ with some properties.

Define $$W^{1,2}[0,\infty) := \{ f \in L^2[0,\infty): f' \in L^2[0,\infty)\}.$$ It is a Hilbert space with the inner product $$ \langle f,g \rangle = \langle f,g \rangle_{L^2} + \langle f',g' \rangle_{...
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1answer
48 views

How are Sobolev spaces on compact Riemannian manifolds defined?

For an open subset $\Omega\subset \mathbb R^n$ one can define the Sobolev space $$H^1(\Omega):=W^{1,2}(\Omega)=\{u \in L^2(\Omega) \, \vert \, \partial u \in L^2(\Omega)\}.$$ Is there a "simple" way ...
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1answer
36 views

Confusion about Sobolev spaces

Reading through Chapter 8 of Brezis. We see that $C_{c}^{\infty}(\mathbb{R})$ is dense in $W^{1,\,p}(I)$. Later on in the chapter we define $W^{1,\,p}_{0}(I)$ to be the closure of $C_{c}^{1}(I)$ in $W^...
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1answer
24 views

Extending Poisson equation from half unit disc

Let $B_+$ and $B$ denote the upper half and full unit disc in $\mathbb{R}^2$ respectively. Suppose $f \in L^2(B_+)$ and that $u \in H_0^1(B_+)$ is a weak solution to the Dirichlet BVP $-\Delta u =f$ ...
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2answers
34 views

How does $\| \nabla u \|_{L^2}^2 \leq C \|\nabla u \|_{L^2}$ imply $\| u\|_{W^{1,2}}^2 \leq \| u\|_{W^{1,2}}$?

So I'm working on some notes and I found this inequality that I really can't make sense of it. This is what is going on: Let $u \in W_0^{1,2}$ and let $f \in L^2$ both on some $\Omega \subseteq \...
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Why are Sobolev spaces useful?

Why are Sobolev spaces useful, and what problems were they developed to overcome? I'm particularly interested in their relation to PDEs, as they are often described as the 'natural space in which to ...
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Is the Sobolev space $W^{1,p}(\Omega)$ always a Hilbert space if $\Omega \subset \mathbb{R}^2$ is bounded and has a continuous boundary?

Well... The title says it all. I know that for $p = 2$ the space $W^{1,p}(\Omega)$ is a Hilbert space, because $L^2(\Omega)$ is. But what if $ 1 < p < \infty$? Is there some theorem or statement,...
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Convergence of Mollifier in $W_0^{l,p}(\Omega)$

I want to prove that for $u \in W_0^{l,p}(\Omega)$ we have convergence of mollifiers $u_\rho \rightarrow u$. I appreciate that $u_\rho \rightarrow u$ in $L^p(\Omega)$ so just need to show convergence ...
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$H^1(\Omega)$ in Euclidean space vs $H^1(\Sigma)$ on a compact surface

For an open subset $\Omega \subset \mathbb R^n$ the Sobolev space $H^1(\Omega)=W^{1,2}(\Omega)$ is defined as \begin{equation} H^1(\Omega)=\{ u \in L^2(\Omega) \, \vert \, \partial^{\alpha}u \in L^2(\...
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Does a function $f \in W^{n,p}(U)$ which vanishes on $U-\overline{V}$ automatically lives in $W^{n,p}_0 (V)$?

Suppose $U,V$ are open sets in $\mathbb{R}^n$ and $\overline{V} \subseteq U$. Suppose $f \in W^{n,p} (U)$ is $0$ on $U-\overline{V}$. Can we say that the restriction of $f$ to $V$ (denoted again by $...
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1answer
21 views

Intersection of all Sobolev spaces with negative order

Call $H^{s}$ the usual $L^2$-based Sobolev spaces on, say, a closed manifold, for $s \in \mathbb R$. The intersection $\bigcap _{s<0} H^s $ contains $L^2$. Is this intersection equal to $L^2$? ...
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56 views

Which space is bigger? (Sobolev Space)

First, let us define the following space : \begin{align*} H^{1}(\Omega) :=\{u :\Omega\to\mathbb{R}\,|\,\forall\alpha\text{ multiindex },0\leq|\alpha|\leq1,||D^{\alpha}u||_{L^{2}(\Omega)}<\infty\} \\...
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1answer
40 views

Estimation in Sobolev norm

I try to understand the last part of Interior Regularity theorem. The theorem says: if $u\in W^{1,2}(\Omega)$ , where $\Omega$ is in $R^n$ and bounded, and $u$ is the weak solution of $\Delta u = f $ ...
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1answer
31 views

Sobolev Inequality in 1D

Reading through Brezis and I am having trouble understand the proof of the following theorem: There exists a constant $C$ such that $\|u\|_{\infty}\leq C\|u\|_{W^{1,\,p}}$ $\forall u\in W^{1,\,p}$, $...
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Dense in Weighted Sobolev Space [on hold]

We know that the $C^\infty$ function space on a bounded interval $I$ of $\Bbb R$ is dense in the Sobolev space on the same domain. My question is: Is the $C^\infty$ function space on a bounded ...
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29 views

Sobolev space extension operator

Let $1\leq p\leq\infty$. There exists a bounded linear operator $P:W^{1,\,p}(I)\rightarrow W^{1,\,p}(\mathbb{R})$, called an extension operator, satisfying the following properties: (i) $Pu|_{I}=u\...
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Contraction semigroup of nonlinear heat equations

Let $(S(t))_{t\geq 0}$ be a semigroup generated by $\Delta$ operator in $L^{2}(\Omega)$ for a bounded smooth domain $\Omega \subset \mathbb{R}^{N}$. Here we assume \begin{align} \begin{cases} D(B) = \{...
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32 views

$W_{loc}^{1,2}$ regularity of nonnegative subharmonic functions

I'm trying to solve the following excercise: Let $v \in C(D)$ be a nonnegative subharmonic function in an open set $D$. Prove that $v \in W_{loc}^{1,2}(D)$ The problem has the following hint: ...
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Density of $C^\infty(\overline{\Omega})$ in $W^{k,2}(\Omega)$

First, $\Omega\subseteq \mathbb{R}^n$ and $$\Omega=\prod_{i=1}^n\left(a_i,b_i\right). $$ I know that the space of test functions $C^\infty_c(\Omega)$ is indeed dense in $W^{k,2}(\Omega)$. Is there ...
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0answers
46 views

Differentiability under the integral sign

Do we have some kind of Lebesgue theorem in the case of an integral defined over a boundary? $\frac{d}{dt}\int_{\partial\Omega}f(t,x)d\sigma$
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26 views

Differentiability of functional

Let $\Omega$ be a bounded domain with Lipschitz border. We define $G:W^{1,p} \rightarrow\mathbb{R}$ by $$G(u)=\int_{\partial\Omega}|u|^pd\sigma.$$ How can I show that $(G'(u),h)=\int_{\partial\Omega}|...
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48 views

Inequality of a p-laplacian

Let $\Omega$ be a bounded domain with lipschitzian boundary.$2\leq p$ Why do we have $$\int_{\Omega}\langle|\nabla u|^{p-2}\nabla u- |\nabla v |^{p-2}\nabla v,\nabla v\rangle\leq \int_{\Omega}(|\nabla ...
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19 views

A boundary equality and dual of the trace's image [duplicate]

Let $\Omega$ be a bounded domain with lipschitzian boundary. How can I show that $\left\langle\frac{\partial u}{\partial n},h\right\rangle_{W^{\frac{1}{p' },p}}=0, \ \forall h\in W^{1,p}\implies \...
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0answers
20 views

Maximal monotone functional

Given a bounded domain $\Omega$ $h:W^{1,p}(\Omega)\rightarrow W^{1,p}(\Omega)^*$ defined by $$ h(u)=\int_{\Omega}|Du|^{p-2}(Du,Dh)_{\mathbb{R}^N}dz$$ why is it continuous and maximal monotone ?
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1answer
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Is $C^k$ function space is dense in Sobolev space? [on hold]

It is known that $C^\infty$ function space dense in Sobolev space. But I want to know that is this statement true also for $C^k$ function space?
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1answer
30 views

Sequentially continuity

Given a map $E:L^\infty\rightarrow C^1(\overline{\Omega}), g_n\rightarrow g$ in $L^\infty(\Omega)$. $u_n=E(g_n), n\in \mathbb{N}$ and u=E(g) and show that every time we take a subsection we can find ...
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0answers
23 views

Inequality involving periodic functions and Sobolev space.

Set $n\in\mathbb{N}$. For $\psi=u+iv$ in $H^1_T(\mathbb{R}^n)=\lbrace \psi\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)$, I wonder if $$ \int_Q |\nabla u|^2\geq C\int_Q (u-1)^3 ...
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0answers
38 views

Evans pde proof of global approximation theorem

I have a question regarding step 3 of the proof (shown in picture below). I am wondering if there is a need for us to consider the subset $V\Subset U$ instead of directly using $U$ when showing the ...
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1answer
29 views

Sequentially lower semicontinuity

Let $\Omega$ be a bounded open set with lipschitz boundary, How can we show that the functional defined by $f:W^{1,p}\rightarrow\mathbb{R}$ $f(u)=\int_{\Omega}|Du|_{\mathbb{R}^N}^p$ is sequentialy ...
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0answers
22 views

Reference request: Density of testfunctions in sobolev space $ W^{1}_{0}$

can someone help me out and name a good source where the following statement ist proven? $C^\infty_0(\Omega)$ is dense in $W^{1,p}_0(\Omega)$ with $\Omega \subset \mathbb{R}^n$ being an open, bounded ...
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0answers
9 views

well-posedness of elliptic pdes with gaussian weights

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure ...