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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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
votes
0answers
169 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 ...
0
votes
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 $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$ ?
0
votes
0answers
37 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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
vote
1answer
30 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
votes
1answer
30 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
votes
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
votes
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. ...
1
vote
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
votes
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
votes
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
votes
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
votes
0answers
31 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
74 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
votes
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
votes
1answer
91 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
vote
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
votes
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}) : ...
1
vote
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 ...
0
votes
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
vote
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
votes
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
votes
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
votes
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 ...
1
vote
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
vote
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
vote
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
52 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 ...
2
votes
1answer
40 views

Critical Homogeneous Sobolev Embedding

For $s\in\mathbb{R}$, $1<p<\infty$, define the homogeneous Sobolev space $\dot{W}^{s,p}(\mathbb{R}^{n})$ as follows: For $f\in\mathcal{S}_{0}(\mathbb{R}^{n})$ (Schwartz functions with Fourier ...
1
vote
1answer
71 views

Poincaré constant for a ball (circle)

I've been recently looking for a best possible Poincaré constant for a particular domains $\Omega$ (it's related to my previous question Unique weak solution to Helmholtz equation on a square) for ...
0
votes
1answer
29 views

Taking $\inf$ for sobolev space in different order

Let $\Omega\subset \mathbb R^N$ open bounded, smooth boundary be given. Define $$ F(u,v):=\int_\Omega |\nabla u|^2v^2dx+\int_\Omega(|\nabla v|^2+(1-v)^2)dx, $$ and two sets $\mathcal U:=\{u\in ...
1
vote
0answers
15 views

compact embedding in sobolev spaces ($W^{1,1}(\mathbb{R}^n)$ in $L^1(\mathbb{R}^n)$ [duplicate]

i have this question : in an example of the compact embedding, the autor gives a demonstration of : the sobolev space $W^{1,1}(\mathbb{R}^n)$ is not compactly embedded in $L^1(\mathbb{R}^n)$ and it ...
2
votes
1answer
21 views

Periodic Poincaré Inequality?

The classical periodic Poincaré inequality states that if $u\in H^1(\mathbb T^n)$ is such that $\displaystyle\int_{\mathbb T^n} u(x)\ dx=0$ then $$\|u\|_{L^2(\mathbb T^n)}^2\leq C_d \|\nabla ...
1
vote
1answer
30 views

Is there $f\in H^1(\mathbb T^n)$ such that $ \textrm{div}(f)=\sum_{j=1}^n \partial_j f=1$?

Is there any $f\in H^1(\mathbb T^n)$ such that: $$\textrm{div}(f):=\sum_{j=1}^n \partial_j f=1,$$ where $1$ stands for the constant function $x\longmapsto 1$. Thanks.
1
vote
1answer
37 views

Inequality for Sobolev fractional spaces

I recall that the Fourier transform of a function $f \in L^1 (\mathbb{R})$ is defined by $$\hat{f}(\xi) = \frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} f(x) e^{- i x \xi} \, dx.$$ We can define that ...
1
vote
0answers
29 views

Variational formulation curl-div equation

I want to prove that the following problem admits a unique solution $A_I\in H(curl;\Omega_I)\cap H(div;\Omega_I)$. $$ \begin{cases} curl(\varepsilon_I^{-1}curl A_I)=curl v_I\;\;\text{in}\,\,\Omega_I\\ ...
2
votes
1answer
16 views

Unique weak solution to Helmholtz equation on a square

I've recently started studying the modern theory of PDEs. I studied some basic properties of Sobolev space and then started with linear elliptic PDEs. I consider the following problem: For which ...