Questions tagged [sobolev-spaces]

For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

Filter by
Sorted by
Tagged with
1
vote
0answers
439 views

Application of method of continuity in partial differential equations

Consider a differential operator $$L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $$\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + \sum_{i,j}\|...
3
votes
0answers
119 views

Truncation in singular integrals

After some thinking, I have a terrible headache caused by the following problem. Imagine we have a function $u \colon \mathbb{R}^n \to \mathbb{R}$ such that $u \in L^2(\mathbb{R}^n)$ and$$\int_{\...
3
votes
1answer
290 views

Properties of Sobolev space $W^{1,p}(\Omega)$ similar with $L^1$ space

I want to show the following statement: if $u\in W^{1,p}(\Omega)$, then $u_-$,$u_+$ and $|u|\in W^{1,p}(\Omega)$ with $$D(u_+) = Du\cdot I_{u>0}\qquad\text{and}\qquad D(|u|)=Du\cdot \text{sign}...
3
votes
1answer
200 views

Why the extension is continuous?

I'm reading M.E.Taylor's PDE, Vol I, Chapter 5 Linear elliptic equations. I have some problem on Proposition 1.7. I will quote it here: Consider the following boundary problem for $u$: $$ \Delta u=0 \...
3
votes
1answer
181 views

Do we have $\varphi=0$ a.e.?

Suppose we have a sequence of smooth functions with compact support: $\{\varphi_n\}\subseteq C_c^{\infty}(\Omega)$, here $\Omega\subseteq \mathbb{R}^n,n\geqslant 2$ is open. Suppose additionally we ...
12
votes
2answers
1k views

$C^\infty_{0}(\bar\Omega)$ dense in $W^{k,p}(\Omega)$: the closure is necessary?

It is a fundamental result of Sobolve space that Let $\Omega \subset \mathbb{R}^d$ or $\mathbb{R}^d_+$, then $C^\infty_{0}(\bar\Omega)$ is dense in $W^{k,p}(\Omega)$. However, in some ...
9
votes
1answer
314 views

Do the two limits coincide?

Let $a$ be a non negative (positive almost everywhere) weight in $L_{loc}^1(\Omega)$, $\Omega\subseteq\mathbb{R}^n$ is open. For $\varphi\in C_c^{\infty}(\Omega)$ define $$ \Vert\varphi\Vert_a^2=\...
2
votes
1answer
201 views

Sobolev space question

Let $s$ be a non-negative integer and $$H^{s}(\mathbb{R}^{n}) = \{f \in L^{2}(\mathbb{R}^{n}) : \frac{\partial^{\alpha}}{\partial x^{\alpha}}f \in L^{2}(\mathbb{R}^{n})\text{ for all $\alpha$ with $|\...
1
vote
2answers
511 views

Function space / inner product

Let $u,v$ be arbitrary elements of a function space $X$ defined on $\Omega \subset \mathbb{R}^n$. Define $$ (u,v)_2 = \int_\Omega \partial_x u \, \partial_x v + \partial_y u \, \partial_y v \:dx $$ ...
3
votes
1answer
101 views

comparison between spaces

There a lot of function spaces and would be nice if somebody can correct me if I am wrong in comparing a few. I want to compare $C^2,L^2,W^{2,2}$ (continuous up to third derivative, Hilbert space of ...
3
votes
0answers
439 views

Why is this functional coercive?

Let $h\in L^2(0,1)$, such that $|\int h \mathrm{d}x|<1$. On the space $H^1(0,1)$, consider $$J(v)=\frac{1}{2}\int_0^1 v'^2\,\mathrm{d}x+\int_0^1\sqrt{1+v^2}\,\mathrm{d}x-\int_0^1hv \,\mathrm{d}x.$$ ...
3
votes
1answer
3k views

weak derivative of a nondifferentiable function

I am reading a book on Sobolev and having trouble understanding a notion of weak derivative. I consider a function $(x-1)^+=\max(x-1,0),x\in[0,2]$, I have a problem at $x=1$, so it is continuous and "...
1
vote
1answer
383 views

Convergence in fractional Sobolev spaces

Consider the space $H^s=H^s(\mathbb{R}^N)$, where $0<s<1$. Take any $u \in H^s$ and any smooth function $\varphi$ such that $\operatorname{supp}\varphi \subset B(0,R)$, for some radius $R>0$. ...
1
vote
2answers
144 views

What is 'imbedding' with Sobolev space and $ L^2 $ space?

I want to know that the meaning of the following. $$ W^{n,1}\textrm{ is continuously imbedded into }L^2$$ Here, $W^{n,1}$ is a Sobolev space.
8
votes
1answer
2k views

Why is the dual space of $H_0^1(\Omega)$ denoted $H^{-1}(\Omega)$?

Why is the dual of the Sobolev space $H_0^1(\Omega)$ denoted $H^{-1}(\Omega)$ ? For a positive integer $k$, $H^k(\Omega)=W^{k,2}(\Omega)$. What is the motivation behind the $-1$ exponent?
3
votes
2answers
132 views

A basic estimate for Sobolev spaces

Here is a statement that I came upon whilst studying Sobolev spaces, which I cannot quite fill in the gaps: If $s>t>u$ then we can estimate: \begin{equation} (1 + |\xi|)^{2t} \leq \varepsilon (...
3
votes
2answers
966 views

Poincare inequality?

Let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^n$. Does there exist $A = A(\Omega)$ with the property that for any $f \in C^\infty(\bar{\Omega})$ with $f = 0$ on $\partial \Omega$, $\...
4
votes
1answer
2k views

“Poincaré” inequality for $H^1$

I have to show the following: Let $U\subset \mathbb{R}^n$ be nice (i.e. bounded, open and boundary of class $C^1$). Further there's a function $$f:H^1(U) \to \mathbb{R}^n$$ which is continuous and ...
4
votes
1answer
517 views

dense subspaces in Hilbert spaces

So far I know, the Schwartz space $S(\mathbb{R}^n)$ is dense in $L^2(\mathbb{R}^n)$ with respect to the usual $\lVert \cdot \rVert_{L^2}$-norm, i.e., for every function $f \in L^2(\mathbb{R}^n)$ there ...
4
votes
1answer
201 views

Not so obvious calculus question

You have $f \in C^\infty([0,1])$ with $f > 0$. Then $\sqrt{f}$ is easily seen to be differentiable . Prove that there exists a constant $C$ independent of $f$ such that: $$\sup_{x\in[0,1]}\left\...
4
votes
3answers
566 views

Total variation of (weakly) differentiable functions

the total variation of a function $u\in L^1(\Omega)$, $\Omega\subset \mathbb{R}^n$, can be defined as $$ \sup \{ \int_\Omega u \; \mathrm{div} g \; dx:\; g \in C_c^1(\Omega,\mathbb{R}^n), \; \lvert ...
4
votes
1answer
547 views

how can I show that $H_0^k(\Omega)=\{u\in H^k(M):\text{supp }u\subset\overline{\Omega}\}?$

Let $\bar{\Omega}$ be a smooth, compact manifold with boundary; we denote the interior by $\Omega$. We can suppose $\bar{\Omega}$ is contained in a compact, smooth manifold $M$, with $\partial\Omega$ ...
5
votes
1answer
2k views

Under which circumstances is the Laplacian compact?

I want to know when the Laplacian is a compact Operator. Do you know some good literature about this topic? For instance, is the Laplacian compact on the Sobolev space $H^2(\Omega)$? Or maybe on the ...
6
votes
1answer
843 views

Sobolev space and equivalence of norms

I'm considering the space $W\{n,p\}[0,1]$ of functions with $n-1$ continuous derivatives $f^{(n-1)}$ is absolutely continuous and $f^{(n)}$ is in $L^p[0,1]$. The usual norm is the sum of the $p$-norms ...
2
votes
1answer
2k views

sobolev space reflexivity

I am having problem with the following 1)Are $H^{1}$ nad $H^{1}_{0}$ a reflexive spaces? 2)If $u_{n} \rightarrow u$ weakly in $H^{1}_{0}$, can I say that it is same as $(\nabla u_{n} , \nabla w) \...
6
votes
1answer
2k views

Schwartz Space is a subspace of Sobolev Space, but how can I show that?

How can I see that $S(\mathbb{R}) \subset H^s(\mathbb{R})$, where the former is Schwartz and the latter is Sobolev space ? This should be obvious according to my notes but unfortunately I can't make ...
6
votes
2answers
3k views

trace of an $H^1$ function is in $H^\frac{1}{2}$

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with smooth boundary. Let $u \in H^1(\Omega)$. I would like a reference for the fact that the trace of $u$ on $\partial \Omega$ is in $H^\frac{1}{...
3
votes
1answer
221 views

Eigenfunction associated to the smallest eigenvalue of an elliptic operator

Let $(T_n)$ be a sequence of elliptic operators defined in $H^2(\Omega)\cap H_0^1(\Omega)$ to $L^2(\Omega)$, with $\Omega$ being a bounded domain with smooth boundary. All of them have a smallest ...
6
votes
3answers
3k views

Equivalent Norms on Sobolev Spaces

Let $u$ be in the Sobolev space $H^2$. Then the standard definition of the norm is $$ ||u||_{H^2}^2 = \sum_{|\alpha| \leq 2} ||D^{\alpha} u||_{L^2}^2. $$ However, I have also seen it defined this ...
1
vote
0answers
73 views

Show properties of elements of $\mathcal{H}^2$

I have problems with my homework. First of all: $\mathcal{H}^2$ means the Sobolev space, right? Because in the script we are using a $W$ for Sobolev spaces and the $\mathcal{H}^2$ can't be found ...
5
votes
1answer
1k views

Open sets and Poincaré's inequality

In many references, Poincaré inequality is presented in the following way : Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...
2
votes
1answer
170 views

Extension domains for $W^{1,2}$

I'd like to have some hints for a problem I bumped into some times ago but I was not able to solve (even if I think the most is done...). Before the problem, let me recall some definitions. Let $\...
15
votes
2answers
818 views

Smoothing a Sobolev function

Let $u \in H^1({\mathbb R}^n)$, $n \geq 2$. Let $\varphi \in C^\infty_0({\mathbb R}^n)$ with $\varphi \geq 0$. Let $\eta$ be a smoothing kernel with $\eta \in C^\infty_0({\mathbb R}^n)$, $\eta \geq ...
3
votes
2answers
227 views

Minimization problem in Sobolev spaces

This is a homework problem and I don't know how to solve it: Consider the following minimization problem on the real-valued sobolev space $H^{1,2}(\Omega)$ with dimension $n=1$ and $\Omega=(0,1)$: $$...
1
vote
2answers
549 views

Question about limits of weakly convergent sequence in $H^1_0(\Omega)$

Let $H = H_{0}^{1}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ whose boundary $\partial\Omega$ is a smooth manifold. We know that the embedding $$H\hookrightarrow L^s(\Omega)$$ is compact for ...
4
votes
1answer
1k views

Two basic questions on PDEs (trace operator and Sobolev space)

I am a bit unsure about the role of the trace operator. I understand that if you have a PDE that is solved by a function $u$ in some Sobolev space, then it's not necessarily defined on the boundary ...
2
votes
0answers
100 views

Extension of Uncertainty Relations to a specific potential in Schrödinger Equation

Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle \...
12
votes
2answers
1k views

Sobolev meets Wiener

Even though the Wiener process (Brownian motion) is continuous, it has no derivative at any point. Does it at least have weak derivatives?
2
votes
2answers
493 views

Weak lower semicontinuity of a functional on Hilbert space?

Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$ If $\{u_n\}\subset H$ is a sequence such that ...
5
votes
2answers
958 views

Does Uniform Boundedness in the Sobolev Space $W^{1,2}$ and Convergence in $L^p$ $(1 \leq p < 6)$ Imply Convergence in $L^6$?

Let $B$ denote the open unit ball in $\mathbb{R}^3$. I want to either prove or disprove that a sequence of functions $u_m$ in the Sobolev space $W^{1,2}(B)$ which is uniformly bounded in the $W^{1,2}(...
17
votes
3answers
1k views

Are polynomials dense in Gaussian Sobolev space?

Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)...
7
votes
1answer
482 views

Global energy conservation in 3D Burgers' equation?

Is the energy $\| u \|^2_{L^2}$ a conserved quantity for the 3D Burgers' equation for smooth solutions that decay rapidly? Finite time singularities can appear, but I am interested in the behavior ...
4
votes
2answers
1k views

Approximate a positive Sobolev function by positive smooth functions

Here is a problem that I have encountered in PDE book several times. But I have never seen a proof of it. I will be very grateful if someone could give me a proof. Question: Let $B$ be the unit ball ...
14
votes
2answers
680 views

Elliptic regularity in Sobolev spaces of negative order

I am having some trouble with Sobolev spaces of negative order. More precisely I am considering the space $W^{-1,p}(\mathbb{R}^2),$ considered as 'the' dual space of $W^{1,q}(\mathbb{R}^2).$ Question ...
3
votes
1answer
172 views

Extend positive function by positive function in Sobolev spaces

This is in connection to this question. I understand the solution, but I want to ask something else regarding the extension of the function. The question is like this: Suppose that $v$ is a ...
7
votes
2answers
3k views

An inequality in the proof of characterization of the $H^{-1}$ norm in Evans's PDE book

I'm reading through the Sobolev Spaces section in Evans's Partial Differential Equations book, and I was stuck on a theorem characterizing the $H^{-1}$ norm. On page 299 Theorem 1 (in the second ...
8
votes
1answer
456 views

Function in $H^1(\Omega)$ which cannot be extended to a greater Sobolev Space

The problem is like this: Consider the open set $\Omega \in \Bbb{R}^2$ by $\Omega=\{(x,y) : 0<x<1, 0<y<x^2 \}$ Is $\Omega$ with Lipschitz boundary? (i.e. the boundary is locally ...
4
votes
0answers
457 views

Adjoint of multiplication operator on Sobolev space

This is sort of an idle question, and I'll admit I didn't think very hard about it. Let $H^1 = H^1(\mathbb{R}^n)$ be the Sobolev space with norm $||f||_{H^1}^2 = ||f||_{L^2(\mathbb{R}^n)}^2 + ||\...
5
votes
2answers
2k views

Continuous representatives in Sobolev Spaces

My question arise from the study of the possible extensions of Rademacher's Theorem to the Sobolev Space $W^{1,p}(\Omega)$, with $\Omega\subset \mathbb{R}^n$. In specific I'm studying the proof of the ...
4
votes
1answer
782 views

Intuition behind Sobolev norm

This morning I was thinking at the following (simple) fact. Let us consider $[0, 1] \to \mathbb{R}$ functions and define a linear functional $$F(u)=u(1)-u(0).$$ $F$ is not continuous on $L^2(0, 1)$ ...