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 the regularity of $u$ necessary to deduce this result? (Evans PDE)

One of the exercises in Evans book on PDEs (at the end of chapter 7) is given as follows: Assume $$u_k\rightharpoonup u\quad\mbox{in}\quad L^2(0,T;H^1_0(U)),$$ $$u_k'\rightharpoonup v\quad\mbox{in}\...
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40 views

About the dual of Sobolev spaces

I'm reading a paper about PDE, in which the author mentioned the space $W^{-1,1}(\Omega)$, does this mean the dual space $W^{1,\infty}(\Omega)^{*}$ ? I hope not. I only know the Sobolev dual space ...
3
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1answer
66 views

Decomposition of measures acting on sobolev spaces

This is a follow-up question to Decomposition of functionals on sobolev spaces. Let $\Omega \subset \mathbb{R}^n$ be a bounded, open set and $\mu \in H^{-1}(\Omega) = H_0^1(\Omega)^*$. Moreover, let $...
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48 views

Sobolev embedding theorem

I am supposed to prove the following: if $U\subset\mathbb{R}^2$ is open, $g\in H^1(U)$ with $\Delta g\in L^2(U)$ and $K\subset U$ is compact, then $||g|_K||_{C^0(K)}\leqslant C ||g||_{H^2(U)}$. The ...
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79 views

Comparison of Sobolev spaces on an open or closed interval

As noted in my previous question, I am currently working through some books on Sobolev spaces. I am struggling to determine whether, given an interval $I=(0,a)$,the Sobolev spaces $W^{m,p}(I)$ and $W^{...
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Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(...
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1answer
48 views

Is this space a Banach space? 2

Consider the set of functions $$\mathcal{B}=\{v\in L^2(0,T;H^1_0(\Omega)): \partial_tv\in L^2(0,T;H^{-1}(\Omega))\},$$ equipped with the norm $$\|v\|_{\mathcal{B}}:=\|v\|_{L^2(0,T;H^1_0(\Omega))}+\|\...
3
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1answer
56 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let $...
1
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1answer
28 views

Prolongement in Sobolev spaces

Let $\Omega$ be an open bounded set of $R^n$, and let $\omega$ be an open subset of $\Omega$ s.t $\overline{\omega} \subset \Omega.$ For $f\in H_0^1(\omega)$, it is known that the extension of $f$ to ...
4
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1answer
86 views

Extending Sobolev functions by zero

I believe that if you have a sufficiently regular (say Lipschitz) bounded domain $\Omega\subset\Bbb R^n$, then you can extend a function $u\in H^1_0(\Omega)$ by zero, and the extension lies in $H^1(\...
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117 views

Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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1answer
52 views

Show for the Hamilton's operator $H$ that $\overline{(H, C_0^{\infty}(\mathbb{R}))} = (H, W_2^2(\mathbb{R}))$ using Fourier transform

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable real-valued function defined on $\mathbb{R}$ bounded with its first derivative. Consider the Hamilton's operator $H$ such that: $$(Hf)...
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1answer
29 views

$C_0^0((a,b)) \subset W_0^{1,2}((a,b))$?

One can easily show that $W_0^{1,2}((a,b)) \subset C^0((a,b))$ for any finite interval $(a,b)$. Intuitively $W_0^{1,2}((a,b))$ should contain more functions than $C_0^0((a,b))$, but how to prove that? ...
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1answer
106 views

Inequality in Sobolev Space Involving Time

In Evans PDE book, I have the next Theorem: If $u \in W^{1,p}([0,T],X)$ then: i) $u(t) = u(s) + \int_{s}^tu'(\tau) d\tau $ for $0\leq s\leq t \leq T$ ii) $\max_{0\leq t \leq T} \| u(t)\|_X \leq C \|...
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1answer
65 views

a nontrivial inequality in the proof of weak solution of biharmonic equation

Hi I am looking at the post discussed about weak solution of biharmonic equation Proving unique weak solution. I am having trouble verifying statement 2: The bilinear operator is coercive, The claim ...
2
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1answer
118 views

Sobolev spaces over closed domains.

I am currently working through books on Sobolev spaces and I notice that these spaces are almost always defined over open domains, i.e. we look at $W^{m,p}(\Omega)$, where $\Omega$ is open. Because ...
0
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1answer
34 views

Differential operators on compact manifolds

First I should apologise if this is a bit of a vague question, but I could not find any references for the explicit construction. I've seen it stated in several places that a differential operator on ...
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1answer
88 views

extension by zero for sobolev functions.

Given some function in $H^1(A)$, and if $B$ is an open subset (A also open) containing $A$, do we get an element of $H^1(B)$ if we just extend by 0? I dont think so, but what would be a simlple ...
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1answer
54 views

Basic question about Sobolev spaces involving time

I'm working with Evans PDE book and I can't understand this: Let $U\subset \mathbb{R}^n$ open and $u\in L^{2}([0,T], H_0^1(U))$ with $u' \in L^2([0,T] , H^{-1}(U))$ and now we consider the ...
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0answers
45 views

Minimizing sequences and topology (direct method)

To show the importance of the choice of the topology for the direct method we have been assigned the following exercise which I've not been able to solve due to my lack of understanding on how strong ...
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112 views

Weak convergence and trace operator

Suppose that $u_j\rightharpoonup u$ in $W^{1,p}(\Omega)$ (notice the weak convergence), with $\Omega\subset \mathbb{R}^3$ regular enough. Let $v_j=Tu_j$, and $v=Tu$, where $T:W^{1,p}(\Omega)\to L^p(\...
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1answer
192 views

To what fractional Sobolev spaces does the step function belong? (Sobolev-Slobodeckij norm of step function)

I'm new to fractional Sobolev spaces and I'm curious about the regularity of some simple functions like e.$\,$g. step functions in order to understand these spaces better. In more detail, for $\Omega ...
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1answer
53 views

Can we say $\| f\|_{\dot H^s(\mathbb R^n)} \le \| f \|_{\dot H^q(\mathbb R^n)}$ if $s\le q$?

If $s\le q$, then can we say that $$ \| f\|_{\dot H^s(\mathbb R^n)} \le \| f \|_{\dot H^q(\mathbb R^n)} $$ holds? Here the homogeneous Sobolev seminorm $\|f\|_{\dot H^s(\mathbb R^n)} = \|(-\Delta)^{s/...
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1answer
45 views

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?
3
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1answer
157 views

Sobolev spaces and the domain of fractional Laplacian

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 $\mathbb{H}^s(\Omega)$. ...
0
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1answer
40 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 x_i}d\lambda^n=-\int_\...
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1answer
63 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 (0,\infty)\...
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0answers
85 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
141 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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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 $\langle\;\cdot\;,\;\cdot\;\...
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1answer
38 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 N}\...
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1answer
374 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 subset of $\mathbb{R}^N$ with $N>p$), is it true that $u_n(x)$ converges to $u(x)$ ...
2
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1answer
47 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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36 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 $$\|u\|_{...
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1answer
50 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
43 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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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 L^{q}(\...
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1answer
55 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
138 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 x_1^...
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239 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 ...
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1answer
46 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 $$\...
3
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0answers
51 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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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
35 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 R^2),\,\|v\|_{L^\infty}\...
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0answers
72 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
115 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
79 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
54 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 then:...
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1answer
60 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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28 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 \|_{L^2(\...