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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1answer
41 views

Inequality Sobolev space

I am stuck with following exercise: Show that for $u \in H^1{(0,1)} = W^{1,p}(0,1)$ denoting the Sobolev space with $p = 2$ and with $u(0) = 0$: $$ \| u\|_{L^\infty{(0,1)}} \leq \| u' ...
1
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1answer
114 views

I want to prove $f\notin W^{1,1}(\mathbb{R},\gamma_{1})$

Let $\gamma_{1}=\mathscr{N}(0,I_{1})$ in $\mathbb{R}$ be the standard Gaussian measure. Consider the sequence $(f_{n})_{n\in\mathbb{N}}\in C_{b}^{1}(\mathbb{R})$ defined by $$f_{n}(x)=\begin{cases} ...
0
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1answer
36 views

The Poisson equation with $L^2$ data has a unique weak solution in $H^1$

In picture below, I have proved that if $f\in L^2(U)$ , (*) has unique weak solution. I use the Lax Milgran to prove it. I think some conditions are redundant.
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0answers
32 views

A Sobolev or Lebesgue space over an unbounded domain will not be separable

Somehow it seems like it should be obvious that a Sobolev or Lebesgue space over an unbounded domain will not be separable since many people state it without explanation. Could someone please ...
1
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1answer
23 views

Equality for functions in $H^2(\mathbb{R})$

I recently stumbled on the following equality: $$ \| (-\frac{d^2}{dx^2} + 1)^{1/2}g\|_{L^2} = \| g\|_{H^1}$$ for $g \in H^2(\mathbb{R})$. I tried to deduce the equality but failed (since I don't ...
2
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0answers
32 views

$H^1_0(M)$ for non-compact $M$

Consider a complete non-compact Riemannian manifold $M$. My question is, is it possible for a non-zero constant function $c$ to be in $H^1_0(M)$? My guess is, this should be possible when $M$ has ...
3
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2answers
27 views

Sobolev embedding for weighted spaces

Suppose that $ p \in (1,2) $, $ \tilde p = \frac{ 2p }{ 2 - p } $ and $ \rho > 1 $. Is the following result true: $$ W_{ \rho }^{ 1, p }(\mathbb{R}^2) \subset L^s_{ \rho }( \mathbb{R}^2 ) $$ for ...
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1answer
15 views

Problem in passage in proof from Willem's book: is that $h$ in $L^{p'}$? How else can I use Dominated Convergence if not?

Here is my problem: Some info: $1\leq p<\infty$; $u\in W^{1,p}(\Omega)$; $\Omega=U\times(-r,r)$, where $r>0$ and $U\subseteq\mathbb{R}^{N-1}$ is open; $\Omega^+=\Omega\cap\{x_N\geq0\}$, ...
2
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2answers
48 views

Obtain $L^p$ norm by using “Riesz Representation”

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in W^{1,2}(\Omega)$ be given. Hence, we have $$ \int_\Omega|{\nabla u}|^2<\infty. $$ Let $\nu\in \mathcal S^{N-1}$ be ...
9
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0answers
172 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in ...
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1answer
31 views

How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$?

$\Omega\subset \mathbb R^n$ is bounded and open. $u,v\in H_0^1(\Omega)$. $Du$ is gradient of $u$. How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$ ?
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0answers
38 views

Equivalence of the $H^1$ norm and the energy entropy norm

Let $D \subset \mathbb R^2$ be a bounded domain with smooth boundary. Let $\mathbf v: D \to \mathbb R^2$ be a divergence free vector field tangent to the boundary, i.e., $\mbox{ div } \mathbf v = 0$ ...
0
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1answer
25 views

Need help understanding proof for a Poincaré inequality.

I need help understanding the first part of the proof for a Poincaré inequality taken from Evans p. 141. Because I couldn't find a proper way to display the average integral symbol, I will be using ...
0
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1answer
32 views

Base of infinity dimensional function space

As I know ,the Sobolev space $H^k(\Omega)=W^{k,2}(\Omega) $ and $L^2(\Omega)$ are Hilbert space. So, they must have orthogonal basis. But I can't find it on my book. Where I can find it ? I In fact ...
2
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0answers
52 views

Strong convergence

I have a sequence $(u_n)$ such that for a functional $I:W^{1,p}_0(\mathbb{R}^N)\rightarrow \mathbb{R}$ of $C^1-$classe we have $$I'(u_n)u_n=0, \forall n\in \mathbb{N}$$ and $$ \nabla u_n (x) \to ...
2
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0answers
23 views

seeking for a general strategy to identify the right space for the domain of semigroup generator

I wish to show the domain of a strongly continuous semigroup $S(t)$ is some sobolev space, for instance, for the heat semigroup, it is known that $$D(A)=W^{2,p}$$ I believe a general strategy to get ...
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0answers
27 views

Chain rule in $\mathbb{R}^d$, with $d\ge 2$.

Given $\Omega\subset\mathbb{R}^d$ be an open bounded set with Lipschitz boundary, let $v\in (H^1(\Omega))^d$, $\psi\in H^1(\Omega)$ $T_K(x):=B_K x+b_K$, where $B_K$ is a non-singular invertible ...
5
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0answers
45 views

How to proof $C_0^\infty(\mathbb{R}^n)$ is dense in $H^s(\mathbb{R}^n)$ by using mollifier

Since the definition of $u\in H^s(\mathbb{R}^n)$ is $\left(1+|\lambda|^2\right)^{s/2}\hat{u}(\lambda)\in L^2(\mathbb{R}^n)$ I find it difficult to give an constructive prove that use mollifier. let ...
0
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1answer
29 views

Sobolev function and polynomials

Let $\Omega\subset\mathbb{R}^2$ be a bounded open set with Lipschitz boundary. Let $v\in H^{l+1}(\Omega)$. I have to prove that There exists a unique $p\in\mathbb{P}_l$ such that $$ \int_\Omega ...
1
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1answer
31 views

the relationship between $W^{k,p}(\Omega)$ and $W^{k,p}_0(\Omega)$

I found the following statement: if $\Omega$ has $C^{\infty}$ boundary, then for all $u\in W^{k,p}(\Omega)$, we can find $u_k\in C^\infty_c(\bar\Omega)$ such that $\|u_k-u\|_{W^{k,p}(\Omega)}\to0$. ...
0
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1answer
32 views

The averages of a subharmonic function over concentric balls increase with radius

Let $B_r$ a ball of radius $r$ in $\mathbb{R}^n$ and $u \in H^{1}(B_r)$ with $\Delta u =0$ in the weak sense. I am reading a paper and the author says that : "since $|\nabla u |^2$ is subharmonic, ...
5
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1answer
39 views

Sobolev embedding into $L^\infty$

I heard that $W^{n,1}(\mathbb R^n)\hookrightarrow L^\infty(\mathbb R^n)$. I can only prove that $W^{1,1}(\mathbb R)\hookrightarrow L^\infty(\mathbb R)$ by Newton-Lebniz formula, how to prove for ...
1
vote
1answer
33 views

Cut off function in N-dimension

Let $\Omega_0, \Omega_1 $ be two open subsets of $R^N, (N\ge 2)$ such that $\overline{\Omega_1} \subset \Omega_0,$ ($\overline{\Omega_1}$ denotes the closure of $\Omega_1).$ I want to know if there ...
6
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0answers
33 views

Sobolev space and integration by parts on non-orientable manifolds

Let $M$ be a compact manifold without boundary which is not orientable. Do all the standard facts that apply to oriented manifolds and Sobolev spaces also apply here? Like Green's formula for example. ...
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0answers
16 views

weak formulation of $u''=\psi'(u)+f$ with $ u\in W^{1,2}_0((a,b))$.

Let $u\in W^{1,2}_0((a,b))$, $(a,b)=I$ and $\psi'$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. Consider $f\in L^2(I)$ and the differential equation $$u''=\psi'(u)+f.$$ I want to ...
2
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0answers
28 views

antiderivative of $\psi'(u)$ for $u\in W^{1,2}_0((a,b))$

Let $u\in W^{1,2}_0((a,b))$ and $\psi'$ the derivative of a convex function $\Psi\in C^1(\mathbb{R})$. If I want to consider the antiderivative of $\psi'(u)$, what happens with the $u$ inside ...
4
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1answer
42 views

The finite product of $L^p$ spaces is reflexive ($1<p<\infty$)

I am trying to understand the proof that the Sobolev Space $W^{1,p}$ is reflexive given in Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. There it is used ...
2
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2answers
35 views

The Sobolev Space $W^{1,p}(I)$ is complete

I am trying to verify that Sobolev Space $W^{1,p}(I)$ is complete. This is the definition of $W^{1,p}(I)$: $W^{1,p}(I)= \{ u\in L^{p}(I) | \exists g \in L^p(I) :\int_{I}^{}u\varphi'= - ...
3
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0answers
32 views

Rellich-Kondrachov compacteness theorem for the Euclidean space with Gaussian measure

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 ...
3
votes
2answers
75 views

Intuition behind the definition of the space $H_{0}^{1}(I) (=W_{0}^{1,2})$

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. It is defined the Sobolev space $W^{1,p}(I)$ as: $$W^{1,p}(I)= \{ u\in L^{p}(I) | \exists g \in ...
4
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1answer
34 views

How to show $ \sup\limits_{k}||u_k||_{W^{1.q}(U)}<\infty $?

If $u_k\rightharpoonup u$ weakly in $W^{1,q}(U)$, how can I show $$ \sup\limits_{k}||u_k||_{W^{1.q}(U)}<\infty? $$
6
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1answer
93 views

The Schwartz function and the sobolev space $W^{2,p}$

How do you prove the Schwartz functions in $\mathbb{R}^n$ are dense in the space $W^{2,p}(\mathbb{R}^n)?$ Terrence tao has a version of the proof of The space $C_c^{\infty}(\mathbb{R}^d)$ of test ...
1
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1answer
38 views

A density argument in the proof of Hardy's inequality for $H^1(\mathbb R^3)$

I understand $C^{\infty}_0 (\mathbb{R}^3)$ is dense in $H^1 (\mathbb{R}^3)$. But I don't understand the reason it is enough to prove the following Hardy inequality if you prove the case $u \in ...
1
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1answer
24 views

If function $u$, is $ C^1$ function on two almost disjoint compact sets, then is $u \in W^{1,\infty}$ of union of two sets?

Let B denote the open unit ball in $R^n$, $B_+ = \{x \in B : x_n > 0\}$ and $B_- = \{x \in B : x_n < 0\}$. Also Suppose u ∈ $C^1(\overline B_+)\bigcap C^1(\overline B_-)$. I was trying to ...
9
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2answers
71 views

$W^{s,p}(\mathbb{R}^{n})$ Is Not Closed Under Multiplication when $s\leq n/p$

For $s\in\mathbb{R}$, $1<p<\infty$, and $n\geq 1$, define the Sobolev space $W^{s,p}(\mathbb{R}^{n})$ by $$W^{s,p}(\mathbb{R}^{n}):=\left\{f\in\mathcal{S}(\mathbb{R}^{n}) : ...
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0answers
26 views

Equivalence of norms on $W^{1, p}(I)$

Let $I=(a, b)$, $a<y<b$ an arbitrary point and $J\subseteq I$ an open interval. Then the norms $\left\Vert \cdot \right\Vert_{\sharp}$, $\left\Vert \cdot \right\Vert_{\flat}$ on the Sobolev ...
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0answers
14 views

compact embedding in sobolev spaces?

i have this question , i want to prove that : if $N>p-1$ the embedding of $W^{1-\frac{1}{p},p}(]0,1[^N)$ in $ L^{\frac{Np}{N-p+1}}(]0,1[^N)$ is not compact. ($p>1$ and real) so how to prove ...
1
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2answers
82 views

If $f,g_j\in\mathcal{C}(U)$ and $\frac{\partial f}{\partial x_j}=g_j$ weakly $\Rightarrow$ $f\in\mathcal{C}^1(U)$

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f,g\in \mathcal{C}(U)$. If $$\frac{\partial f}{\partial x_j}=g$$ for some $j$ $(j=1,\ldots,n)$ in the sense of distributions, how to prove that ...
0
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0answers
42 views

an iff proof on the existence of weak derivative

I have trouble understanding the following proposition. Proposition $f,g\in L_{\text{loc}}^1(\Omega)$. Then $g=D^{\alpha}f$ iff. there exists $f_m\in C^{\infty}(\Omega)$ such that $f_m\to f$ in ...
0
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1answer
14 views

Are there $H^1_0(\Omega)$-functions in the plane that are discontinuous over curves?

Consider a bounded domain $\Omega \subset \mathbb{R}^2$ with Lipschitz-boundary and a curve $\gamma : [0,1] \rightarrow \Omega$. Is it possible to construct a function $f \in H^1_0(\Omega)$ which is ...
6
votes
1answer
64 views

a continuous path between two sobolev functions without increasing energy

This question has been post on MO a week ago. I move it here to get more luck. Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that ...
1
vote
1answer
24 views

Showing properties of a space using dense subsets (soft)

I'm noticing a lot of times during my functional analysis course, that I'm missing some calculus basics (2 years passed since my last class covering this stuff): Especially when working with Lebesgue- ...
3
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1answer
20 views

$u\in W^{1,1}(U) \Rightarrow u^+\in W^{1,1}(U)$ , where $U\subset\mathbb{R}^n$ open

Im trying to show that for an open set $U\subset\mathbb{R}^n$ and a function $u\in W^{1,1}(U)$, also the positive part $u^+$ is in $W^{1,1}(U)$. My idea is the following: Let $E\subseteq U$ defined ...
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0answers
25 views

Confusion about the definition of Sobolev spaces on manifolds

Let $(M,g)$ be a manifold with metric $g$ parametrized by the mapping $S$ and parametric domain $\Omega$. The sobolev space of order one with respect to the $L_2(M)$-norm $H^1_2(M)$ is defined as ...
1
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1answer
39 views

Using Lax-Milgram for linear ODEs

Consider an Sturm-Liouville deferential equation as: $$Lu=(pu')'+qu$$ and differential equation as: $$Lu+f=0$$ where $u(a)=u(b)=0$. We can convert the problem into a Lax-Milgram form for $f\in C[a,b]$ ...
1
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1answer
22 views

gradient, positive and negative parts of a function?

I have this question: Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and $u^+,u^-$ the positive and negative parts of $u$ respectively. Why do we have this equality: $$\int_\Omega \nabla ...
0
votes
1answer
53 views

necessary and sufficient conditions for the existence of solution in the space $W^{k,p}$

I am learning about weak derivatives and sobolev space. In particular I need help to learn the proving strategy/technique. I have trouble proving on how to show a solution belongs to some sobolev ...
1
vote
0answers
41 views

positive and negative parts of a function?

I have this question that I found in a demonstration of a theorem: Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and $u^+,u^-$ the positive and negative parts of $u$ respectively. Why do ...
0
votes
1answer
21 views

A $W^{1,p}$ function that is unbounded on any open subset of $B_1(0)$.

I'm currently studying the properties of Sobolev Spaces in calculus of variations and functional analysis and was wondering if there is a function, that is $W^{1,p}$ but is unbounded on any open ...
2
votes
1answer
23 views

Inequalities for $L^2$ norms of gradients of functions that weakly converge in a Sobolev space

Let $\Sigma$ be a $k$-dimensional compact manifold with boundary. Suppose that $W^{1,k}(\Sigma) \subset L^2(\Sigma)$ is compact and that $\{\phi_j \}$ is a sequence that converges weakly in ...