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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.

6
votes
1answer
222 views

Natural question about weak convergence.

Let $u_k, u \in H^{1}(\Omega)$ such that $u_k \rightharpoonup u$ (weak convergence) in $H^{1}(\Omega)$. Is true that $u_{k}^{+}\rightharpoonup u^{+}$ in $\{u\geqslant 0\}$? You can do hypothesis on $\...
2
votes
0answers
155 views

Compactness of an embedding between weighted spaces

I read somewhere that if: $N\geq 2$ is an integer, $p\in ]1,N[$, $r>N/p$, $m\in L^r(0,a)$ (with $a>0$) and $m>0$ a.e. in $(0,a)$, then the weighted Sobolev space $W^{1,p^\prime} ((0,a),m^{-...
1
vote
1answer
87 views

Functional Analysis, Why this statement true?

Why this statement true? If $f \in C^0([0,1], W^{2,2}(K)) $ then $ f \in C^0 ([0,1] \times K)$. $ K \subset R^n $ , W : Sobolev Space.
2
votes
1answer
55 views

What is the meaning of “approximation” in Sobolev spaces?

For example, I want to know that the statements as below. (1) there is a sequence $\{ f_k \} \subset W^{3,2}$ approximating $f$ in $W^{2,2}$. (2) we can approximate $f$ in $K \subset R^n$ by a ...
3
votes
1answer
689 views

Why do we talk about Trace Operator?

What is the importance of a Trace operator in PDE . Although I have read the Wiki page on this but I am not able to connect it to the aspect of solving PDE's. Particularly why do we define Trace ...
7
votes
1answer
2k views

Does weak convergence in Sobolev spaces imply pointwise convergence?

I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that $\|u_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u_m)\rightharpoonup I$ ...
4
votes
1answer
2k views

Reference of Sobolev space

currently I'm using Krylov's book, while consulting Evans (too many details are left out, for my level). Also, Adams 1975 version has been widely cited. So besides these ones, which book in your ...
3
votes
2answers
872 views

Is the Sobolev space $W^{k ,\infty}$ a Banach algebra?

Some Sobolev spaces are closed under multiplication, making them Banach algebras. My question is whether $W^{k ,\infty}$ is a Banach algebra? Since $L^\infty$ is closed under multiplication, I ...
2
votes
1answer
188 views

Sufficient conditions to hold the following inequality.

If I have an inequality: $\lVert u\rVert_{L^p(R^n)} \le C\lVert\nabla u\rVert_{L^q(R^n)}$ , where $C \in (0,\infty)$ and $u \in C_c^1(R)$, is there a relation between $p, q, n$ such that the ...
1
vote
0answers
209 views

A question about weak lower semicontinuity

Let $\Omega \subset \mathbb{R^{n}}$ be a bounded domain and $u , u_j \in H^{1}(\Omega)$ such that $u_j \rightharpoonup u$ in $H^{1}(\Omega)$ $$ F_{1}(u) = \int_{\{ u > 0\}} \dfrac{1}{2}\langle A_1 \...
2
votes
0answers
101 views

Difference between $C^\infty (U)$ and $C^\infty (\overline U)$

I am learning Sobolev spaces. There seem to be a difference while approximating a function in $W^{k,P}(\Omega)$ by smooth function $C^\infty (\Omega)$ and $C^\infty ( \overline \Omega)$, where we use $...
2
votes
1answer
136 views

A question about the proof in functional analysis

I'm now reading Pazy's book about the semi-group operator. To prove the existence of the solution of KdV equation. He define the Hilbert space $H^s(\mathbb{R})$ $$ \Vert u\Vert_s=\left(\int(1+\xi)^s|\...
-1
votes
1answer
111 views

What is “approximation in a Sobolev Space”? For example,

I want to know the meaning of the statement as below. $$ \text{There is a sequence} \;\{f_k\} \subset W^{3,2}\; \text{approximating}\;\; f \;\;\text{in}\;\; W^{2,2}. $$ Here $ W^{n,m} $ means a ...
2
votes
1answer
572 views

Why are only Sobolev spaces with certain exponents Hilbert Space?

I would like to know why $W^{k,2} (\Omega) $ is a Hilbert space , why is it impossible to define inner product in other Sobolev spaces, ie exponent $\ge2$ . Here $||u||_{W^{k,2} (\Omega)} $ = $(\...
1
vote
1answer
60 views

Prove :If $u(x)\in \displaystyle{H^{l}(R)}$, then $u(x)\in \displaystyle C^{l-1}(R)$

I come across this problem in my functional analysis book.Prove: If $u(x)\in \displaystyle{H^{l}(R)}$, then $u(x)\in \displaystyle C^{l-1}(R)$,$\displaystyle\lim_{x\to|\infty|}D^{\alpha}u=0 ,\...
5
votes
1answer
156 views

The reflexivity of the product $L^p(I)\times L^p(I)$

I am starting to read about the Sobolev spaces $W^{1,p}(I),$ where $I$ is an open interval in $\mathbb{R}.$ In order to establish the reflexivity of $W^{1,p}(I)$ for $p\in ]1,\infty[,$ I need the ...
3
votes
1answer
2k views

Poincaré inequality in unbounded domain

Help me please, how can I show that Poincaré inequality doesn't hold in an unbounded domain? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
0
votes
1answer
125 views

Minimizing a norm to get a solution of a pde

Let $\Omega$ be a regular bounded open subset of $\mathbb{R}^3$. The problem is to solve the following pde: $$\left\{\begin{array}{c c}-\Delta u = u^3 & (\Omega)\\u = 0 &(\partial\Omega)\end{...
4
votes
1answer
376 views

Weak derivatives

How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$. Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
1
vote
0answers
437 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
198 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
179 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 ...
11
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
313 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
508 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
438 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
379 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
142 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
959 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
515 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
557 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
833 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 ...
5
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
215 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 ...