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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orthonormal basis or Parseval frame for Sobolev spaces

Consider the uni-variate Sobolev space of order $m$: $$ \mathcal{W}_2^m=\{ f:[0,1] \rightarrow \mathbb{R}|f,f^{(1)},\dots ,f^{(m-1)}\, \text{are absolutely continuous and}\, f^{(m)}\in L^2\}.$$ It is ...
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20 views

Proving a Sobolev-type ineqauality

Given $I=(0,1)$ and $u\in W^{2,p}(0,1)$ for $p>1$. I am trying to prove that for any $\epsilon>0$, the following hold: $$ \|u\|_{L^\infty(I)}+\|u'\|_{L^\infty(I)}\leq ...
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14 views

Examples on Sobolev Spaces and weak derivatives [on hold]

We know that if ${W^{1,1}((0,1)^{2})}\equiv H^{1}((0,1)^{2})$ and if the weak derivative of $u$ $\epsilon$ ${W^{1,1}((0,1)^{2})}$ satisfies $D_{j}$$u=0$ a.e. in $(0,1)^{2}$, then how to show that ...
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Examples on Weak derivatives and Sobolev Spaces

I know that $u(x)=\log\log(1+|x|^{-1})$ is unbounded function.But how to show that $u$ $\epsilon$ ${W^{1,n}(B(0,1))}$ for $n\geq1.$ I know that the weak derivatives $D_{j}u$ for ...
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12 views

Sobolev spaces of maps between manifolds and the Palais-Smale Condition

I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts: Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean ...
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27 views

Compactness of the inclusion operator $L^2(X,L^2(Y))$ in $H^{-1}(X,H^{-1}(Y))$?

It is well known by Rellich-Kondrasov Theorem that $L^2(\Omega)$ is compactly embedded in $H^{-1}(\Omega)$ for sets of finite measure $\Omega\subset\mathbb{R}^n$. Is it true that the injection of ...
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1answer
42 views

The question regrading to density argument in analysis.

I know the density argument in $L^p$ space, in Sobolev spaces, and even in BV are really sweet in many many cases. However, some times author just work on nice functions and comment by "the rest can ...
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0answers
24 views

Another way to show $\nabla u =0$ a.e. on $u=0$ for $u\in H^1(\Omega)$

This is an exercise on Evans PDE book, Ch5. It provides another way to prove $\nabla u =0$ a.e. on $u=0$ for $u\in H^1(\Omega)$ then the one in Evans & Griapy's book. The statement is as ...
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1answer
32 views

Prove a Poincare-Like Inequality

Here is the question. Given any $\alpha>0$ and $u\in H^1(\Omega)$, $\Omega=B(0,1)$ in $n$ dimensions. Then we have $$\int_\Omega |u|^2 dx\leq C(\alpha)\int_\Omega |\nabla u|^2dx $$ provided that ...
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1answer
17 views

The Relation between Holder continuous, absolutely continuous, $W^{1,1}$, and $BV$ functions

I am trying to find out the relation between those spaces. Take $I\subset R$ on the real line. $I$ can be unbounded. Then I have: We first assume $I$ is bounded. If $u\in C^{0,\alpha}(I)$, for ...
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13 views

Embedding of fractional Sobolev spaces

I have a question regarding fractional Sobolev spaces. Given an open bounded set $\Omega\subset \mathbb{R}^{N}$ (Lipschitz, for instance), $s\in (0,1)$ and $1\leq p<q<\infty$, does the following ...
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1answer
20 views

How to get a smoothing operator from a rapid decreasing function?

From John Roe: Elliptic Operators, topology and asymptotic methods, page 82-83. Let $\mathcal{D}$ be a Dirac operator on the spin bundle $S$, then any section $s\in L^{2}(S)$ has a "Fourier ...
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0answers
25 views

dense subspace of $C(0,T)$

I want to prove that the space H of functions which are continuous in [0,T] with weak derivative in $L^2[0,T]$ and their value in 0 is 0, is dense in the space of continuous functions in [0,T] with ...
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2answers
49 views

Sobolev space on union of two open sets

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and ...
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0answers
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reference request about sobolev space and BV space

I am studying Sobolev Space and BV space by using Leoni's and Evans & Gariepy's book. I was wondering that where can I find some explicit example and some computational question of those space. A ...
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21 views

Question about Sobolev spaces. Controlling Divergence.

I have a question about Sobolev spaces, I think I just need a reference. For $\Omega$ an open and bounded subset of $\mathbb{R}^d$, and $\vec{\Phi}\colon \Omega \to \mathbb{R}^d$ a vector valued ...
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0answers
21 views

Find an operator $Z$ in $H^1(0, \infty)$ with $\langle u,Zv\rangle = \int \bar{u}v dx$

I'm working with operators associated to bilinear forms. What I need to find is a continous, linear operator $T$ defined on $H^1((0, \infty))$ [note that $H^1 = W^{1,2}$ is the Sobolev space] such ...
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0answers
18 views

A function in $W^{1,p}(U)$, $U=B^0(0,1)$

Let $\{r_k\}_{k=1}^{\infty}$ be a countable, dense subset of $U=B^0(0,1)$ and a given function by $$u(x) = \sum_{k=1}^{\infty}\frac{1}{2^k}|x - r_k|^{-\alpha},\,\, x \in U$$ For which values of ...
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1answer
23 views

Counterexample for existence of a minimiser in a variational problem

I'm trying to find an example of a minimisation problem of the form $$ \inf \{ J(u) := \int_{\Omega} f(x)|u(x)| + |\nabla u(x)|^2:\, u \in H^1, \, \int u = 1\}$$ with $\Omega$ an open and bounded ...
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1answer
40 views

Uniformly bounded sequence in Hilbert-Sobolev space

Let $\Omega \subset \mathbb{R}$ be a bounded open set with $C%1$ boundary and $H^1(\Omega) = W^{1,2}(\Omega)$ be the Hilbert-Sobolev space. Let ${u_n}$ be a sequence of functions which are uniformly ...
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finding the solution space of helmholtz equation with mixed boundary?

Let $\Omega\subset \mathbb{R}^2$ be a bounded set and $\Omega' \subset \Omega$ such that $\partial\Omega \cap \partial\Omega' = \emptyset$ $$ \left\{ \begin{align} k^2 u + \Delta u &= 0 \quad ...
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1answer
29 views

Is the embedding $L^2(0,T;H^1) \subset L^2(0,T;L^2)$ compact?

Is the embedding $L^2(0,T;H^1(\Omega)) \subset L^2(0,T;L^2(\Omega))$ compact?
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1answer
30 views

The approximation of BV functions

From Evans & Gariepy 's book, I learned that generally for any $u\in BV(R^n)$, we can find $u_n\in BV(R^n)\cap C^\infty (R^n)$ such that $$ \lim_{n\to\infty} \|u_n-u\|_{L^1(R^n)} = 0$$ and ...
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0answers
62 views

Equivalence of two definitions of weak solution (from a book, I don't understand something!!!!)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
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0answers
42 views

Proving a PDE has a particular weak form (check my proof please!)

Let $u_t - \Delta u = f$ hold in $L^2(0,T;H^{-1})$ for a solution $u \in L^2(0,T;H^1_0)$ with $u_t \in L^2(0,T;H^{-1})$. This means $$\int_0^T \left(\langle u_t(t), v(t)\rangle + \int_\Omega \nabla ...
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Inequality in $H^2$

I have tried to prove this result, but it seems too hard. Need Help. Let $U\subseteq\mathbb{R}^n$ a bounded set with smooth boundary, and the differential operator: ...
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3answers
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Closed Operator on a Sobolev space

I am wondering if the following differential operator $A:D(A)( \subset {\bf{H}}) \to {\bf{H}}$ defined on the sobolev space $\mathbf{H}=H_{0}^{k}(0,1)\times {{L}^{2}}(0,1)\text{ }$ is a closed ...
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1answer
26 views

Density of $C\infty([0,T];V)$ in $W(0,T;V,V)$.

Let $W=\{ u \in L^2(0,T;V) : u_t \in L^2(0,T;V)\}$ where $V$ is a Hilbert space in the Gelfand triple $V \subset H \subset V^*$ and $u_t$ is the weak time derivative. Is it true that ...
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2answers
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Fractional Sobolev space $H^{1/2}(-\pi,\pi)$

Let $H^{1/2}(-\pi,\pi)$ be the space of $L^2$ functions whose Fourier series coefficients $\{c_n\}_n$ satisfy the summability constraint $\sum_n |n| |c_n|^2 < \infty$. Are functions in ...
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2answers
31 views

$G(f) \le \|f\|_{H^s(\mathbb R)},\; s>2 \Rightarrow G(f) \le \|f\|_{H^2(\mathbb R)}$?

If a quantity of a function $f$, call $G(f)$ satisfies $$ G(f) \le \|f\|_{H^s(\mathbb R)} $$ for all $s>2$, then can I conclude that this holds for the limiting case $s\to 2$: $$ G(f) \le ...
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38 views

$H^{1/2}$ function but not better

I am looking for an example of a function $f: [0, 1] \longrightarrow \mathbb{R}$ that is in the Sobolev space of order $1/2$, $H^{1/2}([0, 1])$, but not in the Sobolev space of order $1/2 + ...
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1answer
31 views

Estimating the rate of convergence of $|S_Nf-f|$ given that $\|f\|_{H^s}\leq 1$

Given that the Soloblev space norm $$\|f\|_{H^s}^2=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ and the inequality $$\|f(\cdot +\theta)-f\|_{L^2}\leq 2\pi ...
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0answers
28 views

bounding supremum with Sobolev embedding theorem

I consider $d \geq 2$ and the domain $ D = [0,1]^d$. I consider a function $f : D \to \mathbb{R}$ that is $k$ times continuously differentiable. [$k$ can be specified as large as possible.] I am ...
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1answer
37 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
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1answer
62 views

The proof of a Sobolev embedding inequality by a compactness argument

I want to prove the following If $N\ge 3$ there exists a constant $c_0=c_0(\Omega)$ such that for all $\alpha\ge 1$ and $z\in H^1(\Omega)$ \begin{align} ...
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$\left\|f\right\|_{L^{\infty}(\mathbb R^d)} \leq K \left\|f\right\|_{H^{s}(\mathbb R^d)}$

The Swchartz, $\mathcal S(\mathbb R^d)=\left\{f\in C^{\infty}(\mathbb r^d): \sup_{x\in \mathbb R^d}(1+|x|^{2})^{\frac{k}{2}}\sum_{|\alpha\leq l|}|D^{\alpha}f(x)|< \infty\right\}$, for all $k, l \in ...
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1answer
40 views

weak solution of poisson equation

Consider the equation , with $u\in H^1$ $$\begin{cases}\Delta u = f & \text{in }\Omega\\ \displaystyle \frac{\partial u}{\partial \nu} = 0 & \text{in } \partial \Omega\end{cases}$$ where $\nu$ ...
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1answer
52 views

Closure of $C_0^{\infty}$ in $W^{k,p}(\Omega)$

Why is it that in the definition of $W_0^{k,p}(\Omega)$ for $\Omega$ with boundary smooth enough, we only have $D^{\alpha}u$ for all $0\leq|\alpha|\leq k-1$ vanishing at the boundary and not ...
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29 views

Traces of Sobolev functions in an unbounded domain

I have a doubt concerning the trace of Sobolev functions. Let $C=\Omega\times(0,\infty)$ an infinite cylinder of basis a smooth domain $\Omega$ of $\mathbb{R}^{N}$ and consider the classical Sobolev ...
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Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
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1answer
33 views

Two theorems about approximation by smooth functions

Let $U$ be an open subset of $\mathbb{R}^{n}$. The following are two theorems taken from the chapter about Sobolev Spaces of the Evans' book. Theorem 1 Assume $u\in W^{k,p}(U)$ for some $1\le ...
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Sobolev spaces and Lipsschitz continuity [duplicate]

How to show that u $\epsilon$ ${W^{1,\infty}(\Omega)}$ if and only if u is Lipschitz continuous. But I suggested to use the fact that u is Lipshtz means that there is a constant $L>0$ such that ...
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1answer
38 views

Is the sum of two Sobolev spaces defined on two different sets the Sobolev space defined on the union of these two sets?

Is it true that $H^1(\Omega_1 \cup \Omega_2 )=H^1(\Omega_1)+H^1(\Omega_2)$? Below, we already have a counterexample. Let me ask further. If I impose $\Omega_1 \cap \Omega_2=\emptyset$, is there still ...
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1answer
66 views

Is it true that $L^2$ is compactly embedded in $(W^{1,2}_{0})^{\ast}$?

Is it true that $L^{2}(\mathbb R^{n})$ is compactly embedded in $(W^{1,2}_{0}(\mathbb R^{n}))^{\ast}$? If so, how can I prove it? Context I've just started to study Functional Analysis. I tried to ...
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1answer
34 views

Product of weakly differentiable functions

Let $ u,v \in W^{1,1}_\mathrm{loc}(\Omega) $ and assume that $ uv \in L^{1}_\mathrm{loc}(\Omega) $ and $ u\, Dv + v \,Du \in L^{1}_\mathrm{loc}(\Omega) $. I want to prove that $ uv \in ...
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1answer
25 views

Associating a function $v \in H^{k+1}(K)$ with a polynomial function with equal integrals of derivatives

I read the following: For all $v\in H^{k+1}(K)$ we can associate a polynomial $p\in P_k$ (space of polynomial functions with degree $\leq k$), defined by $\forall \alpha \in N^n$, with ...
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1answer
32 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
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2answers
24 views

Constructing a function $u\in (W_0^{1,p}(B)\cap C(B))\setminus C(\overline{B})$.

Let $B\subset \mathbb{R}^2$ be the unit "open" ball with centre in origin. Define a function $u:\overline{B}\to \mathbb{R}$ in the folllowing way: let $(x,y)=(r\cos\theta,r\sin\theta)$ with $r\in ...
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0answers
31 views

Approximating by smooth functions with compact support.

Consider a bounded domain $D \subset \mathbb{R}^n$ and the Sobolev space $H^1_{0}(D):=\overline{C_c^{\infty}(D)}^{W^{1, 2}(D)}$. Further, consider a Sobolev function which happens to be smooth: $u\in ...
3
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0answers
43 views

Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is ...