For questions about or related to Sobolev spaces, which are function spaces equipped with a norm combining norms of a function and its derivatives.

learn more… | top users | synonyms

1
vote
1answer
33 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 ...
1
vote
0answers
47 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
votes
1answer
30 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 ...
1
vote
1answer
30 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
votes
1answer
35 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 ...
0
votes
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 ...
-1
votes
2answers
38 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 ...
1
vote
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 ...
2
votes
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 ...
1
vote
0answers
26 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.
1
vote
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 ...
0
votes
1answer
27 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
votes
1answer
64 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 ...
1
vote
0answers
13 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 ...
1
vote
1answer
74 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.
1
vote
2answers
42 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 ...
0
votes
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 ...
1
vote
1answer
30 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 ...
0
votes
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 ...
3
votes
0answers
42 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 ...
-1
votes
1answer
34 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 ...
2
votes
1answer
61 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 ...
1
vote
1answer
85 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 ...
1
vote
1answer
69 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 ...
0
votes
0answers
31 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 ...
0
votes
0answers
32 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 ...
0
votes
0answers
27 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 ...
0
votes
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
vote
1answer
28 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 ...
1
vote
1answer
53 views

Composition operators on fractional-order Sobolev spaces

Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication provided $s > 1/2$. This is proved for ...
0
votes
0answers
36 views

Evans PDE Chapt 5 problem 4, existenc of smooth functions form a partition of unity

I have to be honest that I am very lost on the same kind of problem about proving existence of smooth function. I have not done much topology, hence unfamiliar with the "covering" business. In the ...
2
votes
1answer
63 views

A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm \begin{equation*} \| u ...
2
votes
1answer
65 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
0
votes
0answers
35 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
1
vote
1answer
28 views

Prove that $\left\{u\in W_0^{1,2}(\Omega):\int_\Omega|u|^{p+1}\;d\lambda^n=1\right\}$ is well-defined and closed

Let $\Omega\subseteq\mathbb{R}^n$ be a domain with a smooth boundary $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $p>1$ such that $$p<\begin{cases}\infty&\text{, if ...
1
vote
1answer
44 views

Are there $L^2$ functions whose Laplacian is in $L^2$ yet its gradient is not in $\mathbf{L}^2$?

Let $\Omega$ be a bounded smooth domain. Is it true that $H_\Delta^1(\Omega):=\{ u\in L^2(\Omega) : \Delta u\in L^2(\Omega)\}\subset H^1(\Omega):=\{ u\in L^2(\Omega) : \nabla u\in \mathbf{L}^2(\Omega ...
1
vote
1answer
40 views

show that $uv\in W^{1,r}(\Omega)$

Let $\Omega\subset\mathbb{R}^{n}$ is bounded,with $\partial\Omega\in C^{1},p,q\geq 1$,$u\in W^{1,p}(\Omega),v\in W^{1,q}(\Omega)$,show that $ uv\in W^{1,r}(\Omega)$.here ...
-2
votes
1answer
42 views

Prove $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$ [duplicate]

Prove that for $n>1$,the non-bounded function $u=\ln\ln\left(1+\frac{1}{|x|}\right)\in W^{1,n}(\Omega)$,Here $\Omega=B(0,1)\subset \mathbb{R}^{n}$ I think we have to prove that $$ ...
1
vote
3answers
69 views

prove that $u$ is equal a.e. to an absolutely continuous function

Prove that if $n=1$ and $u\in W^{1,p}(0,1) $ for some $1\leq p<\infty$, then $u$ is equal a.e. to an absolutely continuous function,and $u'$ (which exists a.e.) belongs to $L^{p}(0,1)$. My ...
2
votes
1answer
48 views

Sobolev space on a closed subset

Hi I have a question about Sobolev spaces. Let $U \subset \mathbb{R}^{d} $ be a bounded open subset and $dx$ be a Lebesgue measure on $U$ \begin{align} W^{1,2}(U):=\left\{u \in L^{2}(U;dx): ...
0
votes
0answers
26 views

Stuck in trace theorem

I am reading Sobolev space in the book Partial Differential Equation by Evan and I do not understand some point in the proof of the trace theorem. Let $U$ is open, bounded and $\partial U$ is $C^1$. ...
-1
votes
1answer
31 views

An ineqaulity involving critical Sobolev exponent

This is related to my previous question An inequality involving Sobolev embedding with epsilon. There I wished to get that, for given a nice bounded domain $\Omega$ in $\mathbb{R}^n$, $\forall ...
2
votes
1answer
94 views

Show that $\frac{\int_\Omega|\nabla u|^2+\int_\Omega\alpha|u|^2}{\int_\Omega|u|^2}$ attains a minimum in $W_0^{1,2}(\Omega)$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $|\;\cdot\;|_p$ be the seminorm $$|u|_p^p:=\int_\Omega|\nabla u|^p\;d\lambda^n\;\;\;\text{for ...
3
votes
1answer
36 views

Is it possible, that the fist two weak eigenvalues of $-\Delta$ in a bounded domain are equal?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $\lambda_1$ be the first weak eigenvalue of $-\Delta$ in $\Omega$ $\varphi_1$ be the weak eigenfunction associated with $\lambda_1$ ...
2
votes
1answer
30 views

If $l_i$ is the first weak eigenvalue of $-\Delta$ in a domain $G_i$ and $G_1\subseteq G_2$, then $l_1\ge l_2$ and equality is possible

Let $\Omega_i\subseteq\mathbb{R}^n$ be a domain $\lambda_i$ be the first weak eigenvalue of $-\Delta$ in $\Omega_i$ It's easy to verify that $\Omega_1\subseteq\Omega_2$ implies $\lambda_1\ge ...
1
vote
1answer
39 views

An inequality involving Sobolev embedding with epsilon

Let $\Omega$ be a nice bounded domain in $\mathbb{R}^n$ and let $2^*=2n/(n-2)$ for $n>2$ or anything finite for $n\leq2$, i.e., the Sobolev exponent. Then, for any $p\leq 2^*$, one has $$ ...
0
votes
1answer
35 views

Is $C_0^\infty(\mathbb{R}_+)$ a dense subspace of $W_0^{1,2}(\mathbb{R}_+)$?

I read that in some lecture notes that the space of $C^\infty$ funtions compactly supported on the positive real line is a dense subspace of the Sobolev space $W_0^{1,2}(\mathbb{R}_+)$. How can one ...
0
votes
0answers
20 views

Develop a concept of weak solvability for $-\langle\nabla,A\nabla u\rangle=f$

Let $\Omega\subseteq\mathbb{R}^n$ a domain $f\in L^2(\Omega)$ $A:\Omega\to\mathbb{R}^{n\times n}$ be Borel-measurable with $A(x)$ is symmetric, for all $x\in\Omega$ $\exists c_1,c_2>0$ with ...
0
votes
1answer
107 views

Is $T$ a compact mapping from $W_{0}^{1,2}\left(\Omega\right)$ into itself? [closed]

Let $\Omega$ be an open bounded subset in $\mathbb{R}^{6}$ and $f$ be in $L^{8}\left(\Omega\right)$. For any $w$ in $W_{0}^{1,2}\left(\Omega\right)$, define $T\left(w\right)$ be in ...
2
votes
0answers
47 views

How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...