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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0answers
26 views

$W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...
2
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1answer
13 views

One-sided smooth approximation of Sobolev functions

I'm currently trying to specialise a rather general variational inequality to known simple examples to check if my assumptions on the problem are plausible. While doing this, I stepped over the ...
0
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0answers
7 views

Prove $\mathscr{F}[(1+|x|^2)^{-s}]\in L^1(\mathbb{R}^d)$

Let $s>0$, show that $\mathscr{F}[(1+|x|^2)^{-s}]\in L^1(\mathbb{R}^d)$. The original goal is to prove that $W^{s,p}(\mathbb{R}^d)\hookrightarrow L^p(\mathbb{R}^d)$ for all $s>0,1\le p\le \...
1
vote
0answers
57 views

Name of a Sobolev/Poincare type inequality

It is a basic analysis exercise to verify that for $u \colon [0,1] \times [0,\infty) \to \mathbb R$, if $u(0,t) = 0$ for all $t$, then $$ ||u(\cdot,t)||_\infty \leq ||u_x(\cdot,t)||_2,$$ where the ...
0
votes
1answer
32 views

Good reference for partitions of unity?

I am reading about Sobolev Spaces and regularity theory of PDEs. The partition of unity lemma, as stated in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations, is as ...
4
votes
1answer
67 views

Explanation of spaces of functions in PDE

Let's consider following equation: The problem $$ \begin{cases} -\operatorname{div}\left( p\left(x\right) \nabla{u} \right) + q(x)u = f \quad\text{... on } \Omega \\ u = h(x) \quad\text{... on } \...
2
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0answers
21 views

Splitting the region and estimating fractional Sobolev norms

I've been reading the paper "On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces" by Maz'ya and Shaposhnikova and struggling with the short style ...
0
votes
0answers
18 views

Showing $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ is a Hilbert space

Let $I$ be an open interval in $\mathbb{R}$. We define $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ with the scalar product of the Sobolev space $H^1(I)$, i.e. $(u,v)=(u,v)_{L^2(I)}+(u',v')_{L^...
1
vote
0answers
59 views

Sobolev Space dual

I'm interested in the dual space of the Sobolev space $H^1(\Omega)$ for $\Omega$ a bounded smooth domain. Of course, because $H^1(\Omega)$ being a Hilbert space, it's dual is isomorphic to itself, but ...
1
vote
1answer
51 views

What is assumed when one write “$\nabla f$” for $f\in L^p(\mathbb{R}^n)$?

What is assumed when one write "$\nabla f$" for $f\in L^p(\mathbb{R}^n)$? Here is a problem I'm dealing with for weeks. For a fixed $a\in\mathbb{R}^n$, define $f_{a,r}:= \frac{1}{|B(a,r)|} \int_{...
2
votes
1answer
77 views

If $f$ satisfies $\int_{\Bbb{R}}\frac{f(x)\varphi(x)}{1+x^2}\,dx=0$ for some test functions $\varphi$, is $f$ zero?

Let $D$ be the set of functions $\Bbb{R} \to \Bbb{C}$ of the form $$ D = \Big\{ \sum_{k=1}^{n} c_k \mathrm{e}^{it_k x} : c_1, \cdots, c_n \in \Bbb{R}, t_1, \cdots, t_n \in \Bbb{R}, n \geq 1 \Big\}. $$...
0
votes
1answer
34 views

Tensor product space dense in $H_0^1$?

Given $C_c^{\infty}((a_i,b_i))$ for $i \in \{1,2\}.$ Is it then true that $$C_c^{\infty}(a_1,b_1) \otimes C_c^{\infty} (a_2,b_2)$$ is dense in $H_0^1((a_1,b_1) \times (a_2,b_2))$? Clearly, by ...
1
vote
0answers
14 views

Bound on Rayleigh quotient on $H^{1}_{\text{per}}[0,1]^{n}$

Let $Q=[0,1]^{n}$. Define a $Q$-periodic function as follows: A function $f:\mathbb{R}^{n}\to\mathbb{C}$ is said to be $Q$-periodic if for any $x\in\mathbb{R}^{n}$ and any $z\in\mathbb{Z}^{n}$, it ...
-1
votes
0answers
23 views

Bound $L^{2}$ norm

I have an $L^{2}$ norm say $||v \ \partial_{xxx} u||^{2}_{L^{2}}$ and $u \in H^{3}$. Can I write $||v \ \partial_{xxx} u||^{2}_{L^{2}} \leq ||v||^{2}_{L^{2}} \ ||u||^{2}_{W^{3,\infty}}$
4
votes
2answers
61 views

Does existence of the second weak derivative of $f\in L^2$ imply existence of the first?

Let's consider a function $f\in L^2(\mathbb{R})$ for which the second weak derivative exists and lie in $L^2(\mathbb{R})$, i.e. there exists $f''\in L^2(\mathbb{R})$ such that for all $\varphi\in C_0^\...
3
votes
1answer
48 views

Find the Minimum of $ F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R)$.

Let $F: BV(\mathbb R) \to \mathbb R$ be a functional defined as: \[ F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R). \] Show that there is no minimum on $W^{1,1}$, but the ...
0
votes
1answer
27 views

Passing from classical formulation to weak formulation for a general PDE

I am reading a paper dealing with a general elliptic PDE that I need to transform from classical formulation to weak formulation: $$\left\{\begin{matrix} - \sum_{i=1}^n \sum_{j=1}^n (a_{ij} u_{x_i})_{...
1
vote
1answer
52 views

Can weak convergence in $V$ imply strong convergence in $H$?

In the proof of the existence of strong solutions of the stationary NSE in the setting of Hilbert spaces, the following argument is made in Constantin and Foias's Navier-Stokes Equations (p60): ...
1
vote
1answer
35 views

Are weak (Sobolev) solutions to a linear ODE a classical ones?

Let $\Omega$ be an open subset of $\mathbb{R}$ and let $L$ be the differential operator $$ Lf = \sum_{k=0}^{n-1} a_k f^{(k)} + f^{(n)}, $$ where $a_k$ are reals. I would like to show that every ...
0
votes
0answers
15 views

Alexandrov Maximum Principle and $W^{2}_p$ estimates

I'm reading an article of N. V. Krylov: About an example of N. N. Ural'tseva and weak uniqueness for elliptic operators, Nonlinear partial differential equations and related topics, 131–144. This ...
1
vote
2answers
135 views

$\Delta u$ is bounded. Can we say $u\in C^1$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Let us say it has a Lipschitz boundary. Consider the Laplacian $\Delta$ in the classical sense. Suppose $\Delta u=\frac{\partial^2}{\partial x_1^...
0
votes
1answer
2k views

proof on Poincare's inequality.

This might be a silly question. So basically, I have proved the Poincare's inequality for $p=1$ case. That is, for $u\in W^{1,1}(\Omega)$, I have $||u-\bar{u}||_{L^1}\leq C||\nabla u||_{L^1}$. Here $...
0
votes
0answers
20 views

Why do degenerate PDEs require weighted Sobolev spaces

Is there a reason that weighted Sobolev spaces are required for degenerate PDEs other than the fact that when one sets up the weak formulation of the PDE the weights are naturally present, so it is ...
0
votes
0answers
30 views

Is tensor product of weighted Sobolev spaces dense?

Let $W^{k}_{2,w_1}(\mathbb{R})$ be a weighted sobolev space with positive continuous weight function $w_1$ for the integrals of the function and its derivatives. Let $W^{k}_{2,w_{1,1}}(\mathbb{R}^2)$ ...
2
votes
2answers
49 views

Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions, $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that $$ \|u\|_{H^s}^2=\sum_{k\in\mathbb Z}(1+k^2)^{...
0
votes
1answer
46 views

How to find out if a function belongs to $H^2$ or $H^1$

I'm beginning with Sobolev spaces and I found out, that $$ H^k = W^{k,2}. $$ I've also seen the following exercise recently: $$ \frac{1}{2}u'' = 1 $$ And here I'm supposed to find out if $u$ ...
0
votes
0answers
29 views

Is it true that $|∇u(x)|^2\chi_\Omega=|\nabla (u \chi_\Omega)|^2$

Let $u\in L^\infty(\Omega)\cap H^1(\Omega)$ with $\Omega$ open, bounded and regular (as you wish) domain of $\mathbb{R}^N$. Is it true that $$ \int_\Omega |\nabla u(x)|^2 \mathrm{d} x= \int_{\mathbb{...
1
vote
1answer
39 views

Scalar property of $ C(\Omega)=\sum_{|\alpha|\leq m}\color{blue}{\big|\Omega\big|^{\dfrac{2|\alpha|-n}{n}}} \int_{\Omega}|D^\alpha f|^2\ dx $

This is closely related to a previous question: Scale invariant definition of the Sobolev norm $\|\|_{m,\Omega}$ for $H^m(\Omega)$ This question focuses on the direct calculation (by change of ...
0
votes
2answers
17 views

Under what conditions does $u$ with $supp(u) \subset \Omega$ belong to $H_0^1(\Omega)$?

The question is pretty simple. Suppose that $\Omega \subset \mathbb{R}^n$ is bounded and that $u \in H^1(\Omega)$. Under what conditions does $u$ with $supp(u) \subset \Omega$ belong to $H_0^1(\...
1
vote
1answer
27 views

Question concerning the proof of regularity for the Laplacian $f \in H^m(\Omega) \Rightarrow u \in H^{m+2}(\Omega) $

I am stuck at the proof of Theorem 9.25 in Haim Brezis' Sobolev Spaces, Functional Analysis and Partial Differential Equations. This theorem deals with the regularity for the Dirichlet Problem for ...
0
votes
1answer
15 views

Generalized Poincaré Inequality on H1 proof

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
2
votes
1answer
30 views

Showing that there exists a sequence that converges weakly in $H_0^1(\Omega)$.

Proof of lemma $9.7$ in Haim Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations argues as follows: For an element $u \in H_0^1(\Omega)$ we define $D_h u= \frac{u(x+h)-u(x)}...
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0answers
10 views

$||D_hu||_{L^2(Q_+)} \leq ||\nabla u||_{L^2(Q_+)}$ for $u \in H_0^1(Q_+)$

I want to show the following statement: Given $u \in H^1_0(Q_+)$ with $supp(u) \subset \{x \in \mathbb{R}^n\ | \ (\sum_{i=1}^{n-1} |x_i|^2)^{1/2}< 1 - \delta, \ 0 \leq x_n < 1-\delta \}$ y $...
0
votes
0answers
54 views

Help understanding the proof of Trace Theorem given in Evans

I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997): Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there ...
0
votes
1answer
19 views

Convergence of the inverse in Sobolev spaces

Assume we have a sequence $f_k$ which converges to $f$ in the Sobolev space $H^p(D)$, where $D\subset\mathbb{R}^N$ ($N\geq 2$) is relatively compact and $p\geq 1$ is an integer. We also assume that $$...
1
vote
1answer
38 views

Reference request: $W^{2,2}$ estimates of elliptic PDE with measurable coefficients

I have some questions on solvability of the following elliptic PDE: in $\mathbb{R}^2$, for $f\in L^2$, $$a^{ij} u_{x^i x^j} +b^i u_{x^i} + c u = f.$$ Here $\{a^{ij}(x)\}_{i,j=1,2}$ is symmetric ...
0
votes
0answers
22 views

How do we show the inequality for $p=\infty$?

How can we show the inequality for $p=\infty$ ? Since $\overline{u} \in W^{1, \infty}(\mathbb{R}^n)$ we have that $\overline{u}'$ exists and $\overline{u}, \overline{u}'$ are essentially bounded. ...
0
votes
1answer
91 views

Construct extension of function

If $u \in W^{3,p}(\mathbb{R}^{+})$ how can we construct the catoptric extension $\overline{u}$ of $u$ in $\mathbb{R}$ (reflection) such that $\overline{u} \in W^{3,p}(\mathbb{R})$ ? EDIT: By setting $...
1
vote
1answer
62 views

Why is the Black-Scholes PDE called degenerate

I am working in Mathematical Finance and know that the Black-Scholes PDE is degenerate at $x=0$ (I assumed that this was because at 0 the convection and diffusion terms vanish and one is left $V_{t} = ...
2
votes
1answer
80 views

$H_m(\mathbb{R}^n)$ , the completion of $C_C^{\infty}(\mathbb{R}^n)$

Theorem: Let $m$ be a positive integer. Then $H_m(\mathbb{R}^n)=\{ u \in D'(\mathbb{R}^n): D^{\alpha} u \in L^2(\mathbb{R}^n), |\alpha| \leq m\}$ $\to ||u||_{H_m}^2=(2 \pi)^{-n} \int (1+|\xi|^2)^m |\...
1
vote
0answers
24 views

Determine if a function belongs to the sobolev space $W^{1,p}(\mathbb{R})$ and not to $L^q(\mathbb{R})$

I don't understand the first conclusion of the user Tomas in the exercise Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$....
1
vote
0answers
24 views

Minimizing the functional $\int (|\nabla u|^2- u^{2}V)$ on the Sobolev space $H^1$

I have a question about a function defined on a Banach space. Let $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$ and $V:\Omega \to [0,\infty]$ a bounded function on $\Omega$. Let $H^{1}(\...
1
vote
2answers
29 views

Can we extend a function $u \in H_0^1(\Omega)$ to $\overline{u} \in H_0^1(\widetilde{\Omega})$ with $\Omega \subset \widetilde{\Omega}$?

Suppose we have $u \in H_0^1(\Omega)$. I want to know if it is always possible to extend it to an open set $\widetilde{\Omega}$ such that $\Omega \subset \tilde{\Omega}$ by using the extension: $$\...
2
votes
1answer
101 views

Show that if $\,u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$, then $\,u \in W_{0}^{1,p}\left(I\right)$.

I want to show the following statement ($1 \leq p < \infty$), for an open interval $I$: If $u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$ then $u \in W_{0}^{1,p}\left(I\right) $. $W^...
1
vote
0answers
25 views

Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer: $$ ...
3
votes
1answer
40 views

If $\{\nabla u_j\}$ is Cauchy in $L^p(\mathbb{R}^n)$ and $\int_{B(0,1)} u_j dx = 0$, does $\{u_j\}$ converge in $L^p_{\text{loc}}(\mathbb{R}^n)$?

Let $1 < p < \infty$. Let $\{u_j\}_{j=1}^\infty$ be a sequence of functions in $W^{1,p}_{\text{loc}}(\mathbb{R}^n)$ such that $\nabla u_j \in L^p(\mathbb{R}^n)$ for all $j$, $\int_{B(0,1)} u_j ...
5
votes
0answers
56 views

Schwartz functions dense?

I want to show that the Schwartz functions are dense in $$\left\{f \in L^2; \int |x|^2 \left|f(x)\right|^2 dx + \int |\xi|^2 \left|\hat{f}(\xi)\right|^2 d \xi < \infty\right\}$$ where the norm ...
0
votes
0answers
29 views

assigning boundary values to a weakly differentiable function

There is a sentence in Evans I cannot justify. The claim made is $u=g$ on $\partial U$ in the trace sense. Why? I understand that $u\in H^1$ implies $u\in W^{1,p}(U)$. But we also need the assumption ...
1
vote
1answer
48 views

Is the completion of $C_0^\infty(\mathbb{R}^n)$ with respect to $\int_{\mathbb{R}^n}| \nabla \varphi|^2dx$ contained in $L^2(\mathbb{R}^n)$?

Equip $C_0^\infty(\mathbb{R}^n)$ with the norm $$ \|\varphi\|^2_1 := \int_{\mathbb{R}^n}| \nabla \varphi|^2dx.$$ Indeed, $\| \cdot \|_1$ is a norm on $C_0^\infty(\mathbb{R}^n)$ because any constant ...