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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1answer
84 views

Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0, \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}^{N}$ is open ...
0
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0answers
22 views

Characterization of the Gradient of a Distribution

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ (without topology) $u:\mathcal D(\Omega)\to\mathbb R$ is called distribution on $\Omega$ $:\...
0
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0answers
15 views

Characterization of a set occurring in the Helmholtz-Hodge decomposition

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $q\ge 2$ Each $f\in L^1_{\text{loc}}(\Omega)$ can be identified with $\langle f\rangle\in\mathcal ...
0
votes
1answer
14 views

Riesz map between Sobolev space and its dual

I'm asked to find the Riesz's map: $$R:H^1_0(\Omega) \rightarrow H^{-1}(\Omega) $$ $$R: u \mapsto F_u \quad \text{s.t.} \quad <F_u,v>_*= (u_F,v)_{H_0^1} \ \ \forall v \in H_0^1 $$ I chose $(u,...
0
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0answers
22 views

Proof of the Sobolev Space chain rule from Kesavan's Book

I put chain rule on the title because that's what I think they are asking here: This is taken from Kesavan's Functional Analysis book, exercise 2.9 Suppose $\Omega_1 $ , $\Omega_2 $ are bounded open ...
2
votes
1answer
27 views

Rellich-Kondrachov

I read an article about the Rellich-Kondrachov embedding theorem in Sobolev spaces: https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem. Nevertheless, when I checked the refererence in ...
2
votes
1answer
61 views

Definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis

The following is the definition of the space $H^s(\mathbb{R}^n)$ in Hunter's Applied Analysis: Here a regular distribution is a tempered distribution $T_f$ such that it is given by $$ T_f(\varphi)=\...
0
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0answers
22 views

Showing that $\log (\log 1+\frac{1}{|x|})$ belongs to $W^{1,p}(\Omega)$ for $p \geq 2$.

I want to show that the function $f$ belongs to $W^{1,p}(\mathbb{R}^n)$ for $p \geq 2$, where $f$ is defined as $$ f(x)=\log\left(\log \left(1+\frac{1}{|x|}\right) \right)$$ Note: This is an example ...
1
vote
1answer
21 views

Simple example of a function which is in $W^{1,p}(\Omega)$ but not in $L^{\infty}(\Omega)$?

I am looking for a simple (intuitive) example of a function $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is an open set and, obviously, $p \leq N$. Sobolev embedding theorem asserts ...
1
vote
0answers
30 views

How can we extend $\operatorname{div}:C_c^\infty(\Omega)^d\to L^p(\Omega)$ to $W_0^{1,\:p}(\Omega)^d$?

Let $d\in\mathbb N$ and $p\ge 1$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open and $$W_0^{1,\:p}(\Omega):=\overline{C_c^\infty(\Omega)}^{\left\|\;\cdot\;\...
2
votes
0answers
14 views

Fourier coefficients of $\;\log\log$

I was curious if there is an effective way to compute (the asymptotic of) the Fourier coefficients of $$ F(x)= \log\log\left(\frac{1}{\left\lvert x\right\rvert}\right) \cdot \chi\left(\left\lvert x\...
1
vote
1answer
20 views

Approximation of bounded Sobolev functions in $L^\infty$

I'm trying to understand the problem of my former question in detail, and the crucial point (at least in my attempts to solve the problem) seems to be the following: Let $\Omega\subset \mathbb R^d$...
2
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0answers
21 views

Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
2
votes
0answers
40 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
votes
1answer
23 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
votes
1answer
13 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 \...
2
votes
0answers
35 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 ...
1
vote
0answers
74 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
0answers
20 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^...
0
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0answers
42 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
71 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 } \...
1
vote
0answers
65 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
52 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
81 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 $$ D = \Big\{ \sum_{k=1}^{n} c_k \mathrm{e}^{it_k x} : c_1, \cdots, c_n \in \Bbb{R}, t_1, \cdots, t_n \in \Bbb{R}, n \geq 1 \Big\}. $$...
1
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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 ...
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 ...
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
30 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})_{...
4
votes
2answers
66 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^\...
0
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0answers
16 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 ...
0
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0answers
22 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 ...
1
vote
1answer
37 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
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
61 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
48 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
30 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
28 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
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(\...
2
votes
1answer
32 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)}...
1
vote
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
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. ...
1
vote
1answer
41 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
1answer
22 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
67 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} = ...
1
vote
0answers
25 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
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0answers
26 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: $$\...
1
vote
0answers
28 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
42 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 ...
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 ...