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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Is there a name for embedding of Lebesgue spaces $L^q(a,b) ⊂ L^p(a,b)$

Let $1 ≤ p ≤ q ≤ ∞$ then $L^q(a,b) ⊂ L^p(a,b)$ where $L^q(a,b), L^p(a,b)$ are Lebesgue spaces Is there a name for this relationship? Is this the Sobolev embedding theorem?
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23 views

About domain of $(-\Delta)^{\frac 12}$, do not follow a paper

I'm reading this paper on arxiv link. So far OK. Now this I don't understand. Take $s=\frac 12$. They say that by density the operator $(-\Delta)^s$ is defined on $\mathbb{H}^s(\Omega)$. ...
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1answer
24 views

Continuously differentiable functions are weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $u\in C^1(\Omega)$. I want to show, that $u$ is weakly differentiable, i.e. $$\int_\Omega\psi\frac{\partial u}{\partial ...
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1answer
24 views

Weak convergence in the Sobolev space and compact embeddedness

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sopolev space $u\in C^0\left(\overline\Omega\times [0,\infty)\right)\cap C^{2,1}\left(\Omega\times ...
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25 views

Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
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2answers
53 views

Deny Lions Lemma

I am working through the finite element book by Ciarlet and am currently looking at the Deny Lion's Lemma (Theorem 3.1.1 p. 115). The Lemma essentially wants to show that $\inf_{p \in P_{k}}\Vert v ...
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21 views

Under some regularity assumptions to the boundary $\partial\Omega$, the first weak eigenfunction of $-\Delta$ in $\Omega$ is also a strong one

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and ...
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1answer
18 views

Convergence of a sequence of Hölder continuous functions with respect to the Sobolev norm

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $W_0^{1,2}(\Omega)$ be the Sobolev space and $C^{2,\alpha}$ be the Hölder space for some $\alpha\in (0,1]$. Suppose $(u_k)_{k\in\mathbb ...
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1answer
99 views

Weak convergence and convergence almost everywhere

If a bounded sequence $(u_n)$ converge weakly to $u$ in $W^{1,p}(\Omega)$ (Where $\Omega$ is an open bounded from $\mathbb{R}^N$ and $N>p$) Have we that $u_n(x)$ converge to $u(x)$ almost ...
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1answer
27 views

Compact Imbedding into $L^{1} (\Omega)$ [closed]

Let $\Omega$ be a bounded subset of $\mathbb R^{d}$ with a Lipschitz continuous boundary.Prove that: the canonical imbedding of $BV(\Omega)$ into $L^{1}(\Omega)$ is COMPACT .
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1answer
32 views

Space between $L^1$ and $BV$?

I am looking for a function space $X_s$ such that this space has following properties: $X_s$ is a Banach space, and has lower semi-continuous properties with respect to $L^p$ strong convergence. I ...
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0answers
29 views

Trace zero not needed for $H^2$ regularity if $V_N\subset H^2$ is finite dim?

Reading Evans and this note after asking this question, I have been thinking about the estimates for interior/global regularity in Evans, 6.3.1, theorem 1, and thoerem 4 in 6.3.2, of the form ...
0
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1answer
40 views

$L^2$ inner product split over sub-domains in $\mathbb{R}^3$

I have a bounded Lipschitz domains $\Omega, \Omega_1, \Omega_2 \subset \mathbb{R}^3$ such that $\overline{\Omega}=\overline{\Omega}_1 \cup \overline{\Omega}_2$ and $\Omega_1 \cap \Omega_2=\emptyset$. ...
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1answer
26 views

Existence of the first weak eigenvalue of the Laplacian in a bounded domain

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $$H:=W_0^{1,2}(\Omega):=\left\{u\in L^2(\Omega):\nabla u\in L^2(\Omega)\right\}$$ be the Sobolev space, where $\nabla u$ denotes the weak ...
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30 views

Proof of Sobolev imbedding theorem in Adams

I am struggling to understand the proof of the Sobolev embedding theorem given in Sobolev Spaces by Adams. Specifically section 4.25 (2003 edition). The aim is to prove $W^{m,1}(\Omega) \to ...
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1answer
24 views

Derivative of highest order is enough for the Sobolev norm?

Thinking about the partial derivative in this question $\Delta u$ is bounded. Can we say $u\in C^1$? of mine, I encountered this post. Equivalent Norms on Sobolev Spaces I wonder if this hold when ...
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2answers
70 views

$\Delta u$ is bounded. Can we say $u\in C^1$?

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set. Let us say it has a Lipschitz boundary. Consider the Laplacian $\Delta$ in the classical sense. Suppose $\Delta u=\frac{\partial^2}{\partial ...
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0answers
155 views

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
0
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1answer
24 views

Prove that each $W_0^{1,2}$-function is weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be open. $u\in\mathcal L^1_\text{loc}(\Omega)$ is called weakly differentiable $:\Leftrightarrow$ $\exists v\in\mathcal L^1_\text{loc}(\Omega;\mathbb R^n)$ with ...
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33 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
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0answers
20 views

The relation of the Homogeneous Sobolev norm and general Sobolev norm

I'm wondering if the inequality $$ \left\| F\right\|_{\dot H^k(\mathbb R^n)} \le C\left\| f\right\|_{L^\infty(\mathbb R^n)} \left\| f\right\|_{\dot H^k(\mathbb R^n)} $$ holds for $k\in[0,10]$ then $$ ...
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1answer
30 views

The intersection of $BV$ space.

Given $\Omega\subset \mathbb R^N$ open bounded smooth boundary. We define $$ TV(u,\alpha):=\sup\left\{\int_\Omega u\operatorname{div}v\,dx,\, v\in C_c^\infty(\Omega;\,\mathbb ...
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0answers
43 views

How to apply Sobolev inequalities?

I'm struggling with an application of Sobolev inequalities in Evans' book. He presents his argument like this: For $4<p<5$ we have $2(p-1)=2(p-4)+6=2(p-4)+ 2^*$ and therefore $$\left( ...
3
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1answer
25 views

Bessel potential space: Proof of completeness

I want to know a proof that the (one-dimensional) Bessel potential space (for $p=2$) $$H^s(\mathbb{R})=\left\{f:\mathbb{R}\to\mathbb{C}:\int_{\mathbb{R}}(1+\lvert \xi\rvert^2)^{\frac{s}{2}}\lvert ...
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1answer
28 views

Sobolev norm in the definition of Sobolev spaces

I've seen the Sobolev space defined as: The Sobolev space $H^k(\Omega)$ is the set of all functions $u \in L_2(\Omega)$ for which the weak derivative $\partial^\alpha u \in L_2(\Omega)$ for all ...
3
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1answer
33 views

Definition of Sobolev space $H^s$ and domain of $-\Delta^s$

The spaces below are on $\partial\Omega$, the boundary of a bounded smooth domain $\Omega$. I read this in the book on page 141. Define $H^2 := \{ u \in L^2 \mid (-\Delta u) \in L^2\}$. And ...
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1answer
15 views

standard mollifier (comparing the definition in Evans and wiki)

Hi I am looking at the definition of standard mollifier $\eta$ in Evans, and the $\eta$ from wiki enter link description here Have a very basic question, is the $\eta$ in Evans also compactly ...
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2answers
36 views

Difference quotients and weak derivatives (Evans 5.8.2 theorem 3) [closed]

Could anyone give a proof on the remark below the theorem? Basically it is problem 11 I think the proof relates to example 19.19 (p.375) in enter link description here But I really do not ...
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0answers
27 views

How to derive this Sobolev-type inequality in $\mathbb R^3$?

Does anyone know a simple way to derive the following inequality for smooth, compactly supported functions in $\mathbb R^3$? $$ \max \,\{ |u(x)| \mid x \in \mathbb R^3 \} \;\leq\; K\|\, Du ...
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0answers
33 views

A quick question in PDE and evaluation of norms, need verification

Given $I\subset \mathbb R^N$ open bounded. Then we define two quantities $$ Q_1=\sup\left\{ \int_\Omega u\,\text{div}\phi\,dx, \,\phi\in C_c^\infty(\Omega;\,\mathbb R^2), \,\|\phi\|_{L^\infty}\leq ...
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0answers
25 views

Example of function in $H^1(U)$ which is not continuous, where $U \subset R^2$ has a smooth boundary. [duplicate]

Does anyone have a nice geometric example of function in $H^1(U)$ which is not continuous, where $U \subset R^2$ and has a smooth boundary. I want something that is easy to remember.
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0answers
19 views

Multiplication by a Cutoff and Convergence in $H^s(\mathbb R^n)$

I'm trying to teach myself some things about Sobolev spaces out of McLean, Strongly Elliptic Systems and Boundary Integral Equations. Exercise 3.14 has me stumped for no reason: Let $K_j \subset ...
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1answer
26 views

Sobolev spaces, extensions and embeddings

I have the following statement whith an argumentation which I do not understand. Fix integers $k,l$ such that $0\leq l\leq k$. Then the identity map on $C^\infty(\mathbb{T}^d)$ extends to the ...
2
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1answer
63 views

If $u_{n}\rightharpoonup u$ in $W_{0}^{1,p}\left(\Omega\right)$ , do we have $u_{n}^{+}\rightharpoonup u^{+}$? [closed]

If $u_{n}\rightharpoonup u$ in $W_{0}^{1,p}\left(\Omega\right)$ , do we have $u_{n}^{+}\rightharpoonup u^{+}$ and $u_{n}^{-}\rightharpoonup u^{-}$ in $W_{0}^{1,p}\left(\Omega\right)$ and vise ...
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0answers
12 views

Sobolev space trace theory on $M \times [0,T]$

Let $M$ be a compact Riemannian manifold without a boundary. I wonder how the trace map $T:H^1(M \times [0,T]) \to H^{\frac 12}(M \times \{0,T\})$ is exactly.. can I split it into two trace maps for ...
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1answer
69 views

Sobolev functions counterexample

Let $A=(0,1)^{d}$.Does anyone have a simple example of a funtion in $H_0^1(A)\cap H^2(A)$ that is not in $H^2_0(A)$? Thanks a lot.
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2answers
39 views

Pdes definition of spaces

I am reading Temam's book Navier Stokes Equations and he defines $E(\Omega) = \left\{u \in L^2\left(\Omega\right), \ \operatorname{div}(u) \in L^2\left(\Omega\right)\right\}$. Later he says that if $p ...
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2answers
29 views

If $u$ is a Sobolev function then $\nabla u = 0$ on $\{ u = c\}$.

There is a result of the form: If $u$ is a Sobolev function on some domain then $\nabla u = 0$ on $\{ x \mid u(x) = c\}$ where $c$ is constant. Can someone point me to a specific reference? I ...
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1answer
28 views

Computing weak derivatives on an open square

I am looking at the computation of weak derivative in the blog https://sunlimingbit.wordpress.com/2012/11/25/one-example-related-to-weak-derivative-2/ For equation (3), I have some confusions. Here ...
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1answer
36 views

Is $W_{0}^{1,p}\left(\Omega\right)$ compactly embedded in $L^{\infty}\left(\Omega\right)$?

Let $\Omega$ be an open bounded smooth domain in $\mathbb{R}^{N}$. Let $p>N$, is $W_{0}^{1,p}\left(\Omega\right)$ compactly embedded in $L^{\infty}\left(\Omega\right)$? In some textbook such as ...
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39 views

How can we show that $u$ as a weak solution has properties $u \in L^{\infty}(\Omega)$ , $ u>0 $

Let $\Omega$ be an open domain in $\mathbb {R^n}$ and $f \in C^{\infty}(\Omega)$ then how can we prove there is a weak solution $u \in ‎‎ W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \cap ...
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1answer
33 views

the equivalence of a absolute value function $|D^2 u|$ in problem 10(b) evans pde chapter 5

Can someone tell me whether $|D^2 u|$ is equivalent of writing $\frac{\nabla u}{|\nabla u|}\, D^2 u$? This relates to the post Integrate by parts to prove this inequality I wasn't sure why ...
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1answer
58 views

$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

I am reading an article and there, author claim that $$‎L(.)=\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$$ is coercive if ‎‎$ ‎0\leq ...
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1answer
79 views

a vector calculus problem when coping with problem 9 chapter 5 evans

Hi I was trying to understand the solution given by Ray Yang in the post question 9 - chap 5 evans PDE It gets to the sort of things I am quite bad at... When I get to the point ...
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1answer
64 views

For which $1\le p\le\infty$ does $u$ belong to $W^{1,p}$(\Omega)$?

Hi could anyone help with a solution for problem 7 Evans PDE chapter 5? I think it is basically about checking which $p$ allows $$\int_{\Omega} |u|^p dx+\int_{\Omega}|Du|^p dx<\infty$$ ? But I ...
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0answers
28 views

non existence of weak derivative evans pde chapter 5 example 2

Hi Im looking at the this very basic example, in proving the non existence of weak derivative. I am confused in the last line $$\cdots\lim_{m\to\infty}\left(\int_0^2 v\phi_m dx-\int_0^1\phi_m ...
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0answers
29 views

Mapping properties of a pseudo-differential operator of negative order

Let $H^s$ denote the Sobolev space on $\mathbb{R}^d$. Let $P$ be a pseudo-differential operator of negative order $-m$ where $m > 0$. Let $P^*$ denote its $H^0$-adjoint. Is $P^* P : H^s \to ...
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0answers
26 views

Closedness of the range of differential operator first order

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j ...
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0answers
28 views

Why L belongs to the dual space $H^{-1}$

I'm studying pde using Evans book. In chapter 6 he introduces second order partial differential operators for example : $L= \sum{D_i(a_{ij}D_j(u))+\sum{b_iD_i(u)+cu}}$ I can't understand why $L \in ...
1
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1answer
27 views

Continuous Sobolev Embedding

Does Sobolev spaces $H^s$ continuously embed into $L^2$? It seems like this is the case from this post https://en.wikipedia.org/wiki/Rigged_Hilbert_space where can i find a list of continuous ...