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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6
votes
0answers
71 views
+50

prove or disprove an inequality on bounds of derivatives for radial functions

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$, and $f \in C^\infty(B)$, where $B$ is the unit ball in $\mathbb{R}^n$. Prove or disprove the following. Given any positive integer $k$, ...
0
votes
0answers
12 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
0
votes
1answer
29 views

If $u \in H^1(0,\infty)$, does $u_x(\infty) = 0$?

Let $u \in H^1(0,\infty)$. Then is it true that $u'(x) \to 0$ as $x \to \infty$? I am wondering by Green's formula/IBP in this setting; do I get something like $$\int_0^\infty u_{xx}v = ...
6
votes
1answer
806 views

Rellich–Kondrachov theorem for traces

Let $W^{1,p}(\Omega)$ be the Sobolev space of weakly differentiable functions whose weak derivatives are $p$-integrable, where $\Omega \subset \mathbb R^n$ is a domain with Lipschitz boundary. Let ...
16
votes
2answers
177 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
4
votes
1answer
84 views

Differentiation in Besov–Zygmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The Besov spaces ...
4
votes
0answers
123 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where ...
0
votes
0answers
34 views

Besov norm in $W^{1,2}(\mathbb{R}^n)$

A well known result on Besov spaces is that $\Lambda_1^{2,2}(\mathbb{R}^n)=W^{1,2}(\mathbb{R}^n)$. One way to define this Besov space (without Fourier transform) is to consider $$ ...
0
votes
1answer
238 views

Functions on Besov spaces and Hölder conditions

Consider $f_1(x)=-1/\log(|x|)$ and $f_2(x)=1/\sqrt{-\log(|x|)}$ around $x=0$. It can be shown, just by contradiction, that the previous functions do not verify any (positive) Hölder condition: $$ ...
6
votes
1answer
276 views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
4
votes
1answer
106 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
2
votes
0answers
38 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
0
votes
0answers
11 views

Approximation of Sobolev functions by Polynomials

If $\Omega$ is star-shaped with respect to a ball, then Dupont and Scott show in "Polynomial approximation of functions in Sobolev spaces" that $$ \inf_{p\in \pi_{k-1}(\Omega)}\|u-p\|_{L^\infty}\leq ...
1
vote
1answer
23 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
0
votes
1answer
799 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
2answers
39 views

is a convex continuous function absolutely continuous

Does a continuous convex function $\mathbb{R} \to \mathbb{R}$ belong to $W^{1,1}_{loc}$ ? thank you.
1
vote
1answer
23 views

Does weak convergence in $W^{1,p}$ imply strong convergence in $L^q$?

Does weak convergence in $W^{1,p}(\Omega)$ imply strong convergence in $L^q(\Omega)$ when $\Omega$ is bounded? If $f_j$ converges weakly to $f$ in $W^{1,p}$, what can we say about the $L^q$ ...
2
votes
1answer
55 views

Related question to : If $u \in H^1(U)$, then $Du=0$ a.e. on the set $\{u=0\}$

This problem was asked in here. I need to ask something. Provide details for the following alternative proof that if $u \in H^1(U)$, then $$Du = 0 \text{ a.e. } \, \text{ on the set } \{u=0\}. $$ ...
1
vote
1answer
42 views

Extension of bounded operators between norm spaces

Let $X$ and $Y$ be two Banach Spaces and $X_1$ be a subspace of $X$. If $T$ is a bounded linear operator from $X_1\to Y$, then, this is an extension of $T$ from $X\to Y$ such that $\|T\|_{X}\le ...
2
votes
1answer
36 views

Showing Sobolev space $W^{1,2}$ is a Hilbert space

I have the Sobolev space $W^{1,2}$ consisting of all continuous functions $f \in L^2(\mathbb{R})$ such that there exists an $f'$ with $f(b) - f(a) = \int_a ^b f'(t) dt$. $W^{1,2}$ has inner product ...
-2
votes
0answers
32 views

Sobolev space's-self adjoint operator [closed]

Let $H=L^2(0,2)$ and these operators $Aw=w^{(4)}, D(A)=H^4(0,2)\cup H^2_0(0,2)$ and $Bv=-(d(x)v')', D(B)=\{v\in H^1_0(0,2)\mid dv' \in H^1(0,2)\}$ where $$d(x)=\begin{cases}1\text{ if }0<x<1 \\ ...
1
vote
2answers
90 views

Definition of Sobolev spaces: Fourier transform of tempered distribution

I consider in "McLean - Strongly Elliptic Systems and Boundary Integral Equations" the definition of the Sobolev space for $s \in \mathbb R$ $$ H^s(\mathbb R^n) := \{u \in \mathcal S^*(\mathbb R^n) ...
5
votes
1answer
90 views

Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with ...
1
vote
1answer
45 views

Example 3, Sobolev space Evans

In the following example p.260 Evans. I think I understand everything except for one calculus fact in the second last equation: $$ \int_{\partial ...
0
votes
0answers
42 views

Elliptic W^{2,p}-estimates for a Neumann problem.

Consider the simplest elliptic-Neumann problem in $\Omega\subset \mathbb{R}^n$: $$ -\Delta u+u=f\quad \text{in } \Omega, \quad \frac{\partial u}{\partial \nu}=0\quad \text{on } \partial \Omega. ...
11
votes
1answer
644 views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
0
votes
1answer
28 views

Sobolev/Lebesgue norm estimates in $\mathbb{R}^3$

I'm currently working on a project in which I have to establish some estimates for some global Sobolev and Lebesgue norms. We know that if we have a bounded domain $\Omega$, then for any $q \leq p^*$ ...
2
votes
1answer
43 views

Construction of a function $u$ such that $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ and $u \not\in W_0^{2,2}(\Omega)$

I'm wondering about an example of a function $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ such that $u \not\in W_0^{2,2}(\Omega)$. Clearly $W_0^{2,2}(\Omega) \subset W^{2,2}(\Omega) \cap ...
3
votes
1answer
22 views

If $u \in H^{\frac 12}(\Omega)$ and $c \in \mathbb{R}$ is $(u-c)^+ \in H^{\frac 12}(\Omega)$ too?

If $u \in H^{\frac 12}(\Omega)$ and $c$ is a constant, is the function $$(u-c)^+ \in H^{\frac 12}(\Omega)?$$ Here $(x)^+$ is $x$ when $x > 0$ and $0$ otherwise. If it were $H^1$ then it is a true ...
4
votes
1answer
54 views

How would I show a functional is linear and bounded?

Using the following results, for any $f \in H^1(a,b)$, $f$ is continuous on $[a,b]$, and therefore, $$ \int_a^b f(x) dx = f(\zeta)$$ for some $\zeta \in (a,b)$. In addition $$f(c) = f(\zeta) + ...
1
vote
1answer
20 views

logarithmic sobolev inequalities

I'm reading a paper about PDE in fluid dynamics, where it used something called logarithmic sobolev inequalities: $$||\nabla u||_{\infty}\leq C||\omega||_{\infty} (1+\ln ||u||_m)$$ Where ...
0
votes
0answers
22 views

Estimates of $L^2$-orthogonal projection in $H^1$ and $H^{-1}$-norm

suppose we have a finite element space $M_k$ of $L^2(\Omega)$ and the orthogonal projector $Q_k$, defined by $(Q_k w,v)=(w,v)$ for all $w \in L^2(\Omega)$ and $v\in M_k$. My aim is prove the ...
1
vote
1answer
27 views

Canonical projection of $W^{1,p}(\mathbb{R}^N)$ onto $W_0^{1,p}(\Omega)$

Suppose we have a bounded domain $\Omega \subset \mathbb{R}^N$ with sufficiently smooth boundary $\partial \Omega$. The Sobolev spaces $W^{1,p}(\mathbb{R}^N)$ and $W_0^{1,p}(\Omega)$ are defined as ...
0
votes
1answer
25 views

Lebesgue integration by parts in Sobolev space $W^{1,2}(\mathbb{R})$

Let $\phi, \psi \in W^{1,2}(\mathbb{R}) \subset L^2(\mathbb{R})$ and we want to integrate by parts the following piece: $$\int_{\mathbb{R}}\phi(x)\psi'(x)dx$$ Supposedly, it should look like this: ...
0
votes
1answer
25 views

How does $\inf_{c \in \mathbb{R}} \lVert u - c \rVert_{L^2} \le \lVert \nabla u \rVert_{L^2}$ imply this inequality?

Let $M$ be a compact Riemann manifold with boundary. I want to know, given the inequalities $$ \vert T u \vert_{H^{1/2} (\partial M)} \le C \lVert \nabla u \rVert_{L^2(M)} + \lVert u ...
2
votes
1answer
28 views

Confusion about inclusions of dual spaces

We have the triple inclusions $H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)$, where the second inclusion is not literal but in the sense of distributions. Related to this answer, why is the ...
0
votes
1answer
28 views

What is the norm of $H^4(0, 1) \cap H_0^2(0, 1)$?

Let $I = (0, 1)$ and $H_0^2(I) = \{u \in L^2(I) : u', u'' \in L^2(I), u = u' = 0 \;\; \text{on} \;\; \partial I\}$. What is the norm of $$H^4(I) \cap H_0^2(I)?$$ $\|u\|_{H^4(I)} = \Big[ ...
1
vote
1answer
44 views

Introducing an operator by a bilinear form

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \int\limits_I u''(x) v''(x) \, ...
0
votes
1answer
29 views

What is the dual of $H^{-1}(\Omega)$?

The dual of $H^1_0(\Omega)$ is defined to $H^{-1}(\Omega)$. But what is the dual of $H^{-1}(\Omega)$? Is it $H^1_{0}(\Omega)$? I am solving a problem which requires me to use the dual of ...
0
votes
1answer
24 views

Holder continuous functions embedded in Sobolev

For simplicity, I will consider $u\in W^{1,p}(\Omega)$, where $1<p<\infty$ and $\Omega\subset\mathbb{R}$ is open and bounded. I am able to show that \begin{equation} |u(x)-u(y)| \leq ...
1
vote
1answer
23 views

Fourier transform on fractional Sobolev spaces

We say that a tempered distribution $f$ satisfies $f \in H^s(\mathbb R)$ for some $s \in \mathbb R$ if $(1+|\xi|^2)^{s/2} \hat f \in L^2(\mathbb R)$. Here, $\hat f$ denotes the Fourier transform of ...
3
votes
0answers
31 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles.

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
5
votes
0answers
85 views

Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
0
votes
1answer
15 views

Showing that a bilinear form is coercive

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \underset{I}{\int} u''(x) v''(x) ...
2
votes
1answer
21 views

About fractional Sobolev space

I'm reading a paper used Sobolev space $H^s(\Omega)$ , I only know the definition of these space when $\Omega=R^n$ which used Fourier Transform, what if $\Omega$ is a bounded open set? And I also ...
2
votes
1answer
24 views

Continuity of functionals on Sobolev space

Let $U$ be a bounded set in $R^n$ and $W^{1,p}(U)$ denote a Sobolev space. Suppose $\{w_n\}\subset W^{1,p}(U)$ converges to $w \in W^{1,p}(U)$. Let $I[w]=\int_U F(Dw,w,x)dx$ for $w\in W^{1,p}(U)$, ...
-1
votes
0answers
22 views

an exercise concerning sobolev embedding

I saw the following exercise (or variations of it) in several books: Let $\Omega=B_1(0)$ and let $1\leq p<N$. Consider the functions $\{u_n\}_{n=1}^\infty$ defined on $\Omega$ by: $u_n(x)= ...
0
votes
1answer
48 views

Convergence of $\partial_{x_j} u(x,t)$ when $u$ converges in $L^2$ norm.

I hope you can help me with this question. We take $u(x,t)\in L^\infty_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R},L^2(M))$, the derivatives $\partial_{x_j} u $ exist and are continuous, i.e ...
1
vote
0answers
21 views

A calculation for an open ball in $\mathbb{R}^N$ and function space.

Let $B_r(x)$ denoting the ball of center $x$ and radius $r>0$. We denote by $\lambda_{1,\,B_\rho(y)}$ the first eigenvalue of $-\Delta$ in $W^{1,\,2}_0\left(B_\rho(y)\right)$ and by ...