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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Can the closure of a geodesic chart of a manifold without boundary (and with bounded geometry) be defined as a compact manifold with boundary?

Let $(M,g)$ be a $d$-dimensional smooth, riemannian manifold without boundary, which is geodesically complete, connected, has positive injectivity radius and a bounded geometry (i.e. all derivatives ...
2
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1answer
19 views

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
2
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2answers
30 views

Boundedness of Volterra operator with Sobolev norm

Consider the subspace of $C^\infty([0,1])$ functions in the Sobolev space $H^1$. I want to know whether the Volterra operator \begin{equation} V(f)(t) = \int_0^t f(s) \, ds \end{equation} is bounded ...
3
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1answer
31 views

Is $C^\infty_0$ dense in $C^\infty$ w.r.t. $\|\cdot\|_{L^p}$ and $\|\cdot\|_{W^{1,p}}$?

Is the space $C^\infty_0(\Omega)$ of smooth functions with bounded support, dense in the set of smooth functions $C^\infty(\Omega)$ with respect to the norms $\|\cdot\|_{L^p}$ and the Sobolev-Norm ...
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2answers
40 views

Characterisation of one-dimensional Sobolev space

I've got some doubts proving that $$H^1_0((a,b))=\{u\in AC([a,b]): u'\in L^2 \text{ and } u(a)=u(b)=0\}:=X.$$Let $$\mathcal A=\{v\in C^2([a,b]):v(a)=v(b)=0\}.$$ $H^1_0((a,b))\subseteq X.$ Let ...
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2answers
66 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...
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1answer
23 views

Is it true that, $x\rho(x/t)\in H^{s}$ for $\rho\in \mathcal{D}(\mathbb R)$ and $s>3/2$?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support; and the Sobolev space $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} ...
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1answer
40 views

Embedding of weighted Lebesgue space on Fourier side into $C^\infty$

Given $g$, a continuous real-valued function defined in $\mathbb R^n$, $K^g:=\{u:e^g \hat{u}\in L^2(\mathbb R^n,(1/2 \pi)^nd \xi)\}$. Thus $K^g$ can be viewed as a Hilbert space with natural norm ...
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32 views

Weakening Sobolev coefficient in an elliptic estimate.

One of the classical estimates for Sobolev norms and elliptic operators is the following: let $L$ be an elliptic operator of order $l$, then there is some $C > 0$ such that for all $u \in H^{s+l}$ ...
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29 views

Alternative derivation of Poincaré inequality

I've been trying to prove the Poincaré inequality via a representation formula for Sobolev functions $u\in W^{1,p}(\mathbb{R}^{n})$, $1\leq p < n$, wlog with compact support. The setting is ...
3
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1answer
37 views

What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
2
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1answer
38 views

Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$ ||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}. $$ Let $\{\Omega_n\}$ be a sequence of open ...
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0answers
20 views

Uniform continuity of weighted Sobolev functions.

I am trying to show an embedding result for a weighted Sobolev space and have come to the following problem: I have a function $f: (0,a] \rightarrow \mathbb{R} $ such that: $f$ is bounded and ...
2
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0answers
64 views

The Sobolev space $H^s(\mathbb R^n)$ is closed under multiplication when $s>n/2$ [duplicate]

Show that if $u,v\in H^s(\mathbb{R}^n)$ for $s>{n\over 2}$, then $uv \in H^s(\mathbb{R}^n)$ and $$ \|uv\|_{H^s(\mathbb{R}^n)} \le C\|u\|_{H^s(\mathbb{R}^n)} \|v\|_{H^s(\mathbb{R}^n)}, $$ the ...
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1answer
32 views

How do the existence and $L^p$ integrability of weak derivative affect the smoothness of Sobolev functions?

In the definition of Sobolev spaces, let us say the space $W^{1,p}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^n$. What contributes more to the smoothness of a function $f$: 1-The fact ...
2
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0answers
27 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
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0answers
47 views

Positive part of $y$ with $y\in L^2(0,T; H_0^1(\Omega))$ and $y'\in L^2(0,T; H^{-1}(\Omega))$

Let $\Omega \subset \mathbb R^n$ be a domain, sufficiently smooth. Let $T>0$. Define the space $W(0,T)$ by $$ W(0,T) = \{ y \in L^2(0,T; H^1_0(\Omega)): \ y'\in L^2(0,T;H^{-1}(\Omega)),\ $$ where ...
3
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1answer
39 views

Rellich's theorem for Sobolev space on the torus

From John Roe: Elliptic operators, topology and asymptotic methods, page 73: Let $H^{k}$ be the Soblev space defined on the torus $\mathbb{T}^{n}$ with the discrete $k$-norm: $$ \langle f_{1}, ...
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40 views

About the image of the trace operator for Sobolev spaces.

Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain. Once every convex function is locally Lipschitz, we have that $\partial\Omega$ is Lipschitz, therefore, the trace operator $T: ...
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1answer
106 views

Sobolev, Holder, Lp spaces continuous and compact embeddings proof

I would like to know if the following proof is fine. I haven't filled in all the detail but please let me know what you think about the basic outline.(I am aware that there are posts which have dealt ...
2
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1answer
41 views

Global bounded solution of $u_{tt}=\Delta u-mu+h$ in the Hilbert space $X=H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$

Let $\Omega$ be an open subset of $\mathbb{R^n}$. Consider the linear wave equation $$\begin{cases} \dfrac{\partial^{2}}{\partial t^{2}}u\left(t,x\right)=\Delta ...
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3answers
82 views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
3
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31 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
5
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1answer
144 views

Spectrum of Laplace operator

How can I prove that the spectrum of the Laplace operator $\Delta: H^2(\mathbb{R}^N)\subset L^2(\mathbb{R}^N)\rightarrow L^2(\mathbb{R}^N)$ is $\sigma(\Delta)=]-\infty,0]$?
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709 views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
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0answers
28 views

Is Sobolev space $H^{s}(\mathbb R),$ for $s>\frac{1}{2},$ closed under point wise multiplication? [duplicate]

We note that, $L^{2}(\mathbb R)$ is not closed under point wise multiplication. Let $s>\frac{1}{2};$ and we define Sobolev space, as follows: $H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb ...
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1answer
94 views

Calculus in an abstract space

This is page 302 of PDE Evans, 2nd edition. DEFINITIONS. $\text{(i)}$ The Sobolev space $$W^{1,p}(0,T;X)$$ consists of all functions $\textbf{u} \in L^p(0,T;X)$ such that $\textbf{u}'$ exists in ...
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38 views

Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?

For $\alpha>0^{[1]}$, let $D^\alpha=(-\Delta)^{\frac{\alpha}{2}}$ be the operator defined on all Schwartz class functions $f$ as follows: $$ \widehat{D^\alpha f}(\xi)=\lvert \xi\rvert^\alpha ...
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1answer
135 views

Is this proposition about $L^2$ functions correct?

Is this proposition correct? Will you please give a contour example if it is wrong? If $J \in L^2(\mathbb{R}) \cap C^1 (\mathbb{R})$, $f \in C^{\infty}(\mathbb{R})$, $|f'|\leq K$, where $K > 0$ is ...
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1answer
54 views

p--laplacian problem

For $f \in W^{-1,p'}(\Omega)$, $f \geq 0$, we definr $v,$ $v_{\epsilon}$, $g_{\epsilon}$ by $$-\mathrm{div}(|Dv|^{p-2} Dv)=f \quad \mbox{in} \mathcal{D}'(\Omega), \quad v \in W^{1,p}_0(\Omega)$$ with ...
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1answer
468 views

An inequality in Evans' PDE

In Section $9.2$ Theorem $5$ of Lawrence Evans' Partial Differential Equations, First Edition the author proves that for a large enough $\lambda$, the equation $$\begin{array}-\Delta u+b(\nabla ...
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1answer
50 views

Hardy's inequality

Hardy's inequality (for integrals, I think) presented in Evans' PDE book (pages 296-297) contains a formula whose notation is substantially different than the conventional estimate presentation of ...
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3answers
90 views

Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
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1answer
49 views

Bounding the integral of a function by the integral of its derivative

I have no idea where to begin for this question, so any help would be greatly appreciated! Let $\Omega$ be a square with side 1. Show that $$\left(\int_{\Omega} v^2 \, dx \right)^{1/2} \leq \left( ...
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1answer
34 views

Trace theorem for Sobolev functions: what is the significance of continuous extension to the boundary?

Why in the proof of the trace theorem in L.C.Evans' book on PDE, he has considered functions in $W^{1,p}(\Omega)\bigcap C(\overline{\Omega})$ when it is enough to choose functions in ...
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1answer
62 views

Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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1answer
22 views

About Sobolev-Poincare inequality on compact manifolds

Let $2^* = \frac{2n}{n-2}$ where $n$ is the dimension of a compact closed manifold $M$. We get from the Sobolev/Poincare inequality the identity $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u ...
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1answer
34 views

Morrey's inequality

From PDE Evans, 2nd edition, page 281: Now \begin{align} \int_0^s \int_{\partial B(0,1)} |Du(x+tw)| \, dS(w) dt &=\int_0^s \int_{\partial B(x,t)} \frac{|Du(y)|}{t^{n-1}} \, dS(y) dt \\ ...
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138 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
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43 views

Does the Gagliardo–Nirenberg interpolation inequality hold on compact closed manifolds?

The Gagliardo–Nirenberg interpolation inequality on a bounded domain is of the form $$|D^j u|_{L^p} \leq C_1|D^m u|_{L^r}^\alpha|u|_{L^q}^{1-\alpha} + C_2|u|_{L^s}$$ where are there restrictions on ...
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24 views

Gagliardo-Nirenberg-Sobolev inequality

There is one step of the textbook's proof that I wish could be clarified. Pages 277-278 of PDE Evans, 2nd edition, says: Integrate this inequality with respect to $x_1$: \begin{align} ...
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48 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
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1answer
478 views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
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0answers
29 views

About a property of weighted Sobolev spaces

If we define the weighted Sobolev norm as $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ where $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is ...
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59 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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1answer
42 views

Trace-zero functions in $W^{1,p}$

This is an excerpt of a textbook's proof for a theorem (Trace-zero functions in $W^{1,p}$), from PDE Evans, 2nd edition, page 275. Next let $\zeta \in C^\infty(\mathbb{R}_+)$ satisfy $$\zeta ...
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1answer
31 views

Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
4
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1answer
43 views

Trace Theorem question

From PDE Evans, page 272. My question is towards the bootom of this post. THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator ...
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1answer
42 views

Extension Theorem

From PDE Evans, 2nd edition, pages 268-270. My question is at the bottom of this post. THEOREM 1 (Extension Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Select a bounded open set $V$ ...
3
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1answer
77 views

Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...