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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4
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1answer
102 views

The existence of minimizer in Sobolev space

Let $B\subset \mathbb R^2$ be a unit ball. let $v\in W^{1,2}(B)$ be given. We know that $0\leq v\leq 1$ and it is possible that $v=0$ on some positive $\mathcal L^2$ measurable set in $B$. Let $w\in ...
4
votes
0answers
38 views

$L^{2}$ convergence of sequence $|u_{j}|^{p}\nabla u_{j}$

Suppose a sequence $\{u_{j}\}$ is bounded in the Sobolev space $H^{2+\epsilon}(\Omega)$, for $\epsilon>0$, where $\Omega$ is say a bounded, $C^{\infty}$ domain. Here, the fractional space $H^{s}(\...
2
votes
1answer
22 views

Product of $W^{1,p}_0$ functions

Let $p>n$, and let $f,g\in W^{1,p}_0(\mathbb{R}^n)$ be two sobolev functions. Prove that $fg\in W^{1,p}_0(\mathbb{R}^n)$. I was able to prove the Leibniz formula for weak derivativatives, but ...
0
votes
0answers
45 views

weak derivative of sign function

How the weak derivative of the sign function $$\begin{equation*} sign(x)=\left\{ \begin{array}{rl}1 & \text{if } x> 0,\\ 0 & \text{if } x=0, \\ -1 & \text{if } x<0 \\ \end{array}\...
1
vote
2answers
70 views

Sobolev space definitions

By definition of the Sobolev space $W^{m,p}$ we have : $$W^{m,p}(\Omega)=\{u\in L^p(\Omega)\ |\ \forall \alpha \text{ such that } |\alpha|\le m, D^{\alpha}u\in L^p(\Omega)\}$$ Can someone give me a ...
2
votes
1answer
34 views

Sobolev inequality with a constant independent of the support of the function

I am revising our PDE module and I came across the following version of the Sobolev inequality: $$ \exists C>0 \forall u\in W^{1,p}_0(\mathbb{R}^n):\, \|u\|_{L^{p*}}\le \|Du\|_{L^p}\, , $$ where $p*...
1
vote
1answer
21 views

Lax Milgram, prove continuity for $a : {W^{2,2}(I)} \times {W^{2,2}(I)} \rightarrow\mathbb{R}$

Let $I = (0,1)$ and $b>0$. Let $f\in L^2(I)$. I need to show that there exists a unique $u \in W^{2,2} (I)$ such that $$a(u,\phi) = \int^1_0 (u''\phi'' + bu'\phi' + u\phi) dx = \int^1_0 f\phi \ ...
1
vote
1answer
25 views

Inner Product on Sobolev Space with p=2

Wikipedia defines the Sobolev Space: $H^{s,p}(\mathbb{R}^n)= \left\{f \in L^p(\mathbb{R}^n): \mathcal{F}^{-1}[(1+|k|^2)^{\frac{s}{2}} \mathcal{F}f] \in L^p(\mathbb{R}^n) \right\}$ Where $s \in \...
2
votes
0answers
23 views

Lebesgue Space/Bochner Space interpolation Theorem

I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain, $$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{p_2}(I;L^{p_2}(\Omega))\hookrightarrow L^...
1
vote
1answer
46 views

Linear functional

For $a<b \in \mathbb{R}$, let $G=(a,b)$ be a bounded interval. For every $x \in G$, let the generalized function $\delta_x$ be defined by $$ \int_G \delta_x \phi(x)dx = \phi(x) ~~ \text{for every} ~...
0
votes
1answer
12 views

Inequality with $W^{1,p}$ norm

For $a<b \in \mathbb{R}$, let $G = (a,b)$. How can I show that firstly for every $v \in W^{1,p}(G)$ there exists a unique $\tilde{v} \in C^0(\overline{G})$ such that for almost every $x \in G$ it ...
0
votes
0answers
9 views

Inequality with $W^{1,p}$-norm

How can I show that for every $p \in [1,\infty)$ and every $v \in C^1_c(\mathbb{R})$ that $$ \lvert g_p(v(x)) \rvert \leq p \lVert v \rVert^p_{W^{1,p}(\mathbb{R})}$$ for every $x \in \mathbb{R}$ where ...
0
votes
0answers
22 views

Convergence of Sobolev function in a domain with a curve removed

Let $B\subset \mathbb R^2$ be given as a unit ball. Let $\omega\subset W^{2,2}(B)\cap L^\infty$ be given. (So we are only in 2d space) Let $\Gamma\subset B$ be a closed Lipschitz curve such that $\...
2
votes
1answer
21 views

Alternative characterisation of weak derivatives

Let $\Omega \subseteq \mathbb{R}^n$. The textbook I am reading defines the space $W^{k,2}(\Omega)$ as follows: An element $u$ is in $W^{k,2}(\Omega)$ if there exists a sequence $(u_m)$ in $C^{\...
0
votes
1answer
32 views

The minimum of two Sobolev function

Let $u$, $v\in W^{1,2}(\Omega)$ be two non-negative sobolev functions. We define $$ w:= \begin{cases} u&\text{ if }u\leq v\\ v&\text{ if }v\leq u \end{cases} $$ Let $$ P:=\{x\in\Omega,\, u\...
1
vote
1answer
33 views

The level set of Sobolev function

Let $u\in W^{1,2}(\Omega)$ where $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary. Moreover, we have $0\leq u\leq 1$. Let set $\Gamma$ be defined as $$ \Gamma:=\{x\in\Omega,\,\, u(x)=0\} $...
0
votes
0answers
25 views

Is, on a compact set, the space of analytic functions dense in $W^{k,p}(\mathbb{R}^n)$ (the Sobolev space with non-integer k)

Is there any way to prove that the space $\mathcal{A}$ of functions that are analytic on a compact $\mathcal{K}$ of $\mathbb{C}$ is dense in the space of functions that are in $W^{k,p}(\mathbb{R}^n)$ ...
2
votes
1answer
77 views

Sobolev spaces of sections of vector bundles

Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset \...
0
votes
1answer
52 views

Soblolev space question

I want to prove that if $u(x)$ is in $H^s(\mathbb R^n)$ then $u(cx+d)$ is also in $H^s(\mathbb R^n)$ for any constant $c$ and $d$ .Can someone help me?Thanks
0
votes
1answer
23 views

Inequality involving harmonic functions over the ball and half ball

Let $B\subset \mathbb R^2$ be a unit ball. Let $B^+:=B\cap \{x_2\geq 0\}$ where we set $x=(x_1,x_2)\in \mathbb R^2$. Let $\omega\in C^1(\partial B)$ be given such that $|\nabla \omega|>0$ for all $...
1
vote
0answers
33 views

References for Laplace Equation, weak solutions and eigenvalues via the variational method

Could you point me towards books and/or papers which treat the weak form of Laplace (Poisson) equation and determine the eigenvalues and eigenfunctions of the Laplacian using the variational method (...
5
votes
1answer
139 views

Explicit characterization of dual of $H^1$

Let's start by some well-known facts: $H^1(\mathbb{R})$ is a Hilbert space, hence there holds the Riesz representation theorem, stating that any linear functional on it can be represented as $L = \...
0
votes
1answer
30 views

Sobolev space $H_0^m$ and sobolev norm and seminorm

I have problems understanding the definitions of $H_0^m(\Omega)$-spaces. What does the $0$ stand for? Does it mean that the functions are zero at $\partial\Omega$? Or does it mean that it is non-zero ...
1
vote
0answers
26 views

Prove generalisation of Sobolev embedding theorem using induction.

I am trying to prove the following; I am doing it by induction and the case $k=1$ is already done. So suppose the above is true for all integers less than or equal to k, first we want to show that ...
0
votes
0answers
14 views

The regularity of harmonic function over the tour

Let $B_1,B_2\subset \mathbb R^2$ be the ball centered at $0$ with ridus $1$ and $2$, respectivily. Define $\Omega:=B_2\setminus B_1$. Let $\phi\in C_c^\infty(\mathbb R^2)$. We consider the following ...
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0answers
32 views

Why are Sobolev spaces $W^{m,p}$ denoted with the letter “W”?

Why are the Sobolev spaces $W^{m,p}$ where $m\in\mathbb{Z}$ and $p\in\mathbb{Z}$ with $1\le p\le \infty$ denoted with the letter "$W$"? I know they are denoted with the letter $H$ when $p=2$ because, ...
1
vote
1answer
37 views

Difference between $\{u\in L^2(\Omega)^n\mid \nabla\cdot u\in L^2(\Omega)\}$ and $H^1(\Omega)^n$

Let $\Omega$ have the segment property. Define $$ E(\Omega)=\{u\in L^2(\Omega)^n\mid \nabla\cdot u\in L^2(\Omega)\}, $$ where $L^2(\Omega)^n=L^2(\Omega;\mathbb{R}^n)$ and the derivatives taken in the ...
3
votes
1answer
65 views

Does the convergence of integrals against $H^1_0$ functions imply boundedness in $L^2(\Omega)$?

In order to show something, I would like to have this strange side result, which seems obvious yet surprisingly I cannot find a way to show it "rigoriously". Suppose we have a sequence $(u_n) \subset ...
1
vote
1answer
17 views

Is $H^1$ subspace of Sobolev space $W^{1,1}$?

Let $I \in \mathbb R^d$. A result I need, states that a certain property holds weakly in $BV(I)$, and holds strictly in $W^{1,1}(I)$ (which is a subspace of $BV(I)$). I would actually need this ...
0
votes
0answers
14 views

Do we have $(1+|y|^{-s})(1+|y|^s)\hat{u} \in L^2(\mathbb{R}^n)$?

For $0 < s < \infty, s\in\mathbb{R}$, and $u \in L^2(\mathbb{R}^n)$. Suppose $(1+|y|^s)\hat{u} \in L^2(\mathbb{R}^n)$, do we have $(1+|y|^{-s})(1+|y|^s)\hat{u} \in L^2(\mathbb{R}^n)$? The "hat" ...
0
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0answers
19 views

estimate the energy by the boundary value.

Let $\Omega\subset \mathbb R^N$ be given, where $\Omega$ is open bounded with smooth boundary. Let $\omega\in W^{1,2}(\Omega)$, hence $T[\omega]$, the trace, is well defined. Now define $$ u:=\...
1
vote
1answer
30 views

Extending Functions in Sobolev Spaces

If $U\subset W$ then every function in $L^p (U)$ can be extended to a function in $L^p (W)$, for example by setting it to be 0 outside of $U$. However, not every continuous or differentiable ...
3
votes
0answers
36 views

Steps in alternative proof that if $u \in H^1(\Omega)$, then $Du = 0$ a.e. on set $\{u = 0\}$

Let $\Omega$ be an open subset of $\mathbb{R}^n$, and let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \...
1
vote
1answer
45 views

Identity having to do with weak derivative

For $a<b \in \mathbb{R}$, let $(a,b) = G \subset \mathbb{R}$ be a bounded interval in the real numbers. Show that there exists no $v \in L^2(G)$ and no $y \in G$ such that $$ \int_G v \varphi \text{...
0
votes
0answers
43 views

An execise regarding Neumann problem

I have this exercise, dealing with Neumann problem, i.e. $U$ is a bounded domain in $\mathbb{R}^n$, $-\nabla^2 u=f$ in $U$ $\partial u/\partial n=0$ on $\Gamma$ ($\Gamma=\partial U$) (a) For a given ...
2
votes
2answers
85 views

Composition of a weakly convergent sequence with a nonlinear function

Let $\Omega\subset\mathbb{R}^n $ be bounded smooth domain. Given a sequence $u_m$ in Sobolev space $H=\left \{v\in H^2(\Omega ):\frac{\partial v}{\partial n}=0 \text{ on } \partial \Omega \right \}$ ...
1
vote
1answer
37 views

An equality about the Sobolev space $W^{1,2}$

From the Plancherel identity, we know that $$\int_{\mathbb{R}} |f(x)|^2\,dx=\int_{\mathbb{R}}|\widehat{f}(\xi)|^2\,d\xi$$ is valid for all $L^2(\mathbb{R})$ functions and in particular for Schwartz ...
3
votes
1answer
38 views

Understanding Meyers-Serrin theorem: about the use of mollifiers.

I have read through the Meyers-Serrin theorem, and would like to understand why a simpler argument would not work. The theorem states that $C^{\infty}(\Omega)$ is dense in $W^{k,p}(\Omega), 1 \le p &...
1
vote
1answer
52 views

Poincaré's Inequality on Sobolev Spaces in One Dimension

The following is a version of Poincaré's inequality: Let $I$ be a bounded interval, then there exists a constant $C$ dependent on $I$ such that $$\|u\|_{W^{1,p}(I)} \leq C\|u'\|_{L^p(I)} \ \ \ \ \...
0
votes
0answers
31 views

Example in Sobolev Space

The function $\sqrt{x}$ is in $W^{1,1}(0,1)$, but can it be extended to $W^{1,1,}(-1,1)$? And in particular, is the function $$f(x) = \chi_{[0,1]}\sqrt{x} $$ in $W^{1,1}(-1,1)$? I'm trying to think ...
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0answers
37 views

Elliptic regularity

Given the problem, $f \in L^2(\Omega)$, $a$ is continuous, coercive (elliptic) and bilinear, $L$ is continuous suppose this problem has a unique solution $$\begin{cases} u \in V, \\ a(u,v) = L(v), \...
2
votes
0answers
24 views

Hilbert space and traces

Let $\Omega$ be the open unit ball in $\mathbb{R}^n$, and $\Gamma := \Omega \cap \{x_n=0\}$. Let $\Omega_1 = \{ x \in \Omega: x_n > 0 \}$ and $\Omega_2 = \{ x \in \Omega: x_n < 0 \}$. Define $...
1
vote
1answer
48 views

$W^{1,\infty}(\mathbb{R})$ is the same as $C^{0,1}(\mathbb{R})$

Let $f\in (C_c(\mathbb{R}))^*$ be a distribution. Show that $f\in C^{0,1}(\mathbb{R})$ if and only if $f\in L^\infty(\mathbb{R})$, and the distributional derivative $f'$ of $f$ also lies in $L^\infty(\...
3
votes
1answer
47 views

PDE problem, for every $f \in L^2(0, 1)$, does problem admit a unique solution $u \in H^2(0, 1)$?

Assume that $p \in C^1([0, 1])$ with $p(x) \ge \alpha > 0$ for all $x \in [0, 1]$ and $q \in C([0, 1])$ with $q(x) \ge 0$ for all $x \in [0, 1]$. Let $v_0 \in C^2([0, 1])$ be the unique solution of$...
5
votes
3answers
62 views

Uniqueness of solution $u \in H^2(0, 1)$ to partial differential equation.

Assume that $p \in C^1([0, 1])$ with $p(x) \ge \alpha > 0$ for all $x \in [0, 1]$ and $q \in C([0, 1])$ with $q(x) \ge 0$ for all $x \in [0, 1]$. Let $v_0 \in C^2([0, 1])$ be the unique solution of$...
1
vote
1answer
73 views

Counterexample of Sobolev Embedding Theorem in $W_0^{1,p}$

I am looking for a counterexample of Sobolev Embedding Theorem, i.e. I am seeking for a sobolev function $u\in W_0^{1,p}(\Omega),\,p\in[1,n)$, $\Omega$ is a bounded domain in $\mathbb{R}^n$ such that ...
4
votes
2answers
51 views

Exist unique $g_0 \in H^1(0, 1)$ such that $f(0) = \int_0^1 (f'g_0' + fg_0) \text{ for all }f \in H^1(0, 1)$?

The mapping $f \mapsto f(0)$ from $H^1(0, 1)$ into $\mathbb{R}$ is a continuous linear functional on $H^1(0, 1)$. Does there exist a unique $g_0 \in H^1(0, 1)$ such that$$f(0) = \int_0^1 (f'g_0' + ...
4
votes
1answer
48 views

Unbounded operator, when is it dense?

Let $E = L^p(0, 1)$ with $1 \le p < \infty$. Consider the unbounded operator $A: D(A) \subset E \to E$ defined by$$D(A) = \{u \in W^{1, p}(0, 1),\text{ }u(0) = 0\} \text{ and }Au = u'.$$My two ...
3
votes
0answers
27 views

Characterization of the function $u(x) = (1 + |\log x|)^{-1}$ [closed]

Let $u(x) = (1 + |\log x|)^{-1}$. Are the following all true? $u \in W^{1, 1}(0, 1)$ $u(0) = 0$ ${{u(x)}\over x} \in L^1(0, 1)$
1
vote
0answers
38 views

Relationship between norm of a function and its supremum

Let $u \in H^{1}(\Omega)$ , $0<\tau <1$ and $B_{\tau R}$ the open ball of radius $\tau R$ under what conditions (I) $||u||_{L^{p}(B_{\tau R})} = \sup\limits_{B_{R}} u$ and (II) $||u||_{L^{p}(...