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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3
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1answer
22 views

Duals of Sobolev Spaces vanishing on parts of the boundary

I am revising for a Finite Elements course and have the following question about the definition of $H^{-1}$. Let $D\subseteq \mathbb{R}^2$ be bounded Lipschitz domain and let $\Gamma_0 \subseteq ...
0
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0answers
22 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
3
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1answer
32 views

Double integral definition of periodic Sobolev spaces

Preliminaries: For $s \geq 0$ define the periodic Sobolev spaces $H^s(\mathbb T)$ with norm $$ \| f \|_{H^s(\mathbb T)}^2 = \sum_{k \in \mathbb Z} \big(1 + |k|^2 \big)^s \, \big|\widehat{f}_k \big|^2, ...
-2
votes
0answers
25 views

Sobolev space exercise1 [on hold]

Let $B_{1}(0) \subseteq \mathbb{R}^{n}$ and $f(x)=|x|^{\gamma}$ with $\gamma >0$, what $\gamma $ verified that $f \in W^{1,p} (B_{1}(0))$?
2
votes
1answer
30 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...
1
vote
1answer
34 views

Simple Inequality for Proving Equivalent Besov Seminorms

For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty$, and $h\in\mathbb{R}^{n}$, define the quantity $$I_{p}(h):=\left(\int_{\mathbb{R}^{n}}\left|f(x+h)-f(x)\right|^{p}dx\right)^{1/p}$$ and define ...
1
vote
1answer
24 views

Continuous function not sobolev

Let $I=(a,b)$ an open bounded interval. It is well known that $W^{1,p}(I)\subset C(I)$. It easy to see that there are $f\in C(I)$ such that $f\notin W^{1,p}(I)$ It is enough to take $I=(0,1)$ and ...
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0answers
23 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
2
votes
1answer
29 views

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 ...
0
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0answers
20 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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0answers
47 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 ...
0
votes
0answers
21 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 ...
0
votes
0answers
56 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 ...
1
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0answers
26 views

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 ...
2
votes
1answer
37 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 ...
3
votes
1answer
49 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
vote
1answer
24 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 ...
3
votes
1answer
42 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 ...
5
votes
0answers
94 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 ...
0
votes
1answer
28 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: ...
0
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1answer
22 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? ...
0
votes
1answer
28 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 ...
1
vote
1answer
26 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
votes
1answer
73 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
votes
1answer
21 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 ...
0
votes
1answer
28 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 ...
1
vote
1answer
27 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
30 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 ...
2
votes
0answers
46 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 ...
-1
votes
0answers
30 views

Equivalence of the Sobolev norm $\| f\|_{W^{k,\infty}}$ and $\| f\|_{H^{k,\infty}}$.

I want to know that when $k=0,1,2,...$, the Sobolev norm $$ \| f\|_{W^{k,p}} :=( \sum_{|\alpha|\le k} \|\partial^\alpha f\|_{L^p(\mathbb R^n)}^p )^{1/p} $$ is equivalent to the norm $$ \|f\|_{H^{k,p}} ...
0
votes
1answer
42 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 ...
1
vote
1answer
42 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)} = ...
1
vote
1answer
29 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
votes
1answer
40 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
votes
1answer
30 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 ...
1
vote
1answer
28 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 ...
3
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0answers
34 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
60 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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0answers
26 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 ...
1
vote
1answer
19 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 ...
0
votes
1answer
102 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 ...
2
votes
1answer
37 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 ...
0
votes
0answers
30 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
votes
1answer
41 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 $\Omega_1 \cap \Omega_2=\emptyset$. ...
0
votes
1answer
28 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 ...
4
votes
0answers
35 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 ...
1
vote
1answer
25 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 ...
1
vote
2answers
85 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 ...
6
votes
0answers
161 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
votes
1answer
26 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 ...