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 sum of two Sobolev spaces defined on two different sets the Sobolev space defined on the union of these two sets?

Is it true that $H^1(\Omega_1 \cup \Omega_2 )=H^1(\Omega_1)+H^1(\Omega_2)$? Below, we already have a counterexample. Let me ask further. If I impose $\Omega_1 \cap \Omega_2=\emptyset$, is there still ...
2
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1answer
55 views

Is it true that $L^2$ is compactly embedded in $(W^{1,2}_{0})^{\ast}$?

Is it true that $L^{2}(\mathbb R^{n})$ is compactly embedded in $(W^{1,2}_{0}(\mathbb R^{n}))^{\ast}$? If so, how can I prove it? Context I've just started to study Functional Analysis. I tried to ...
2
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1answer
31 views

Product of weakly differentiable functions

Let $ u,v \in W^{1,1}_\mathrm{loc}(\Omega) $ and assume that $ uv \in L^{1}_\mathrm{loc}(\Omega) $ and $ u\, Dv + v \,Du \in L^{1}_\mathrm{loc}(\Omega) $. I want to prove that $ uv \in ...
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14 views

associating a function v \in H^{k+1}(K) with a polynomial function

for all $v\in H^{k+1}(K)$ we can associate a polynomial $p\in P_k$ (space of polynomial functions with degree $\leq k$), defined by $\forall \alpha \in N^n$, with $|\alpha|\leq k,\ \int_K ...
2
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1answer
20 views

An inequality using Sobolev norms

Let $\| \cdot \|_{H^s(\mathbb R)}$ be the usual Sobolev norm in $\mathbb R$ and $r>0$. If we have $$ \|f\|_{L^\infty(\mathbb R)} \le \| f\|_{H^k(\mathbb R)} $$ for all $k>r$, then the inequality ...
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2answers
23 views

Constructing a function $u\in (W_0^{1,p}(B)\cap C(B))\setminus C(\overline{B})$.

Let $B\subset \mathbb{R}^2$ be the unit "open" ball with centre in origin. Define a function $u:\overline{B}\to \mathbb{R}$ in the folllowing way: let $(x,y)=(r\cos\theta,r\sin\theta)$ with $r\in ...
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0answers
27 views

Approximating by smooth functions with compact support.

Consider a bounded domain $D \subset \mathbb{R}^n$ and the Sobolev space $H^1_{0}(D):=\overline{C_c^{\infty}(D)}^{W^{1, 2}(D)}$. Further, consider a Sobolev function which happens to be smooth: $u\in ...
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0answers
35 views

Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is ...
2
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1answer
48 views

If $u\in W_{0}^{1,p}(\Omega)\cap C(\Omega)$ and $N$ is a nodal domain of $u$ , then does it follow that $u\in W_{0}^{1,p}(N)$?

Assume that $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and that $u\in W_{0}^{1,p}(\Omega)\cap C(\Omega)$ . Let $N\not=\emptyset$ be a nodal domain of $u$ , that is, a connected component ...
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1answer
17 views

Weak derivatives equals zero

Im just learning sobovel space, I was wondering if the weak derivate holds similar things of the original derivate. Let $U\subset \mathbb{R^n}$ is a open set, and $u\in W^{1,p}$ if $$Du=0 \ \ a.e$$ ...
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1answer
25 views

Sobolev embedding counterexample

I trying to find a counterexample to show that $$W^{1,p}(\mathbb{R ^n}) \nsubseteq C^{0,\alpha}(\mathbb{R^n}) $$ for $p>n$ and $\alpha > 1 -\frac{n}{p}$. No clue yet, thanks for your help.
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26 views

Differentiation in Besov-Zigmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The besov spaces ...
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1answer
23 views

Smoothing effect for weak solutions of heat equation

Let $u_0 \in L^2$ and $f \in L^2(0,T;H^{-1})$ and consider the solution $u \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ of $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some BC (eg. zero Dirichlet). I am ...
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0answers
18 views

Property of intersections of Bochner spaces

My question: Assume I have a function $ u \in H^2(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega))$. Now I want to bound the gradient of $u$. Can I deduce that $u \in H^1(0,T;H^1(\Omega))$ and under which ...
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24 views

Canonical injection from $H^1(K)$ in $L^2(K)$ is compact? $K$ compact and connected

Where can i find a proof that the canonical injection from $H^1(K)$ in $L^2(K)$ is compact? $K$ compact and connected
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29 views

When is the second derivative of a $H^{1}$ function in $L^{2}$?

Is there a characterisation of all functions $\phi\in H^{1}$ such that for given functions$\{g^{ij}\in L^{\infty}\}$ then $\sum_{ij} g^{ij}\phi,_{ij}\in L^{2}$ where $\phi,_{ij}=\frac{\partial ...
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1answer
31 views

tempered distribution and sobolev spaces

The Schwartz space $\mathcal S(\mathbb R^d)$ is the set of all complex-valued function $f \in C^{\infty}(\mathbb R^d)$ such that $\sup_{x\in \mathbb R^d}|x^{\alpha}D^{\beta}f(x)|<\infty$ where ...
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1answer
33 views

Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces. We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big ...
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0answers
27 views

Is Sobolev regularity propagated under evolution?

Given a well-posed initial problem in a domain $\Omega$ of the form: \begin{equation} \square\phi=f \end{equation} where $\square$ is the D'Alamebrt operator, $f\in L^{2}(\Omega)$, with initial ...
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0answers
48 views

Characterization of $H^{-1}(U)$

I am trying to understand the proof of characterization of $H^{-1}(U)$ as in Evan's Partial Differential equations. My questions is different to that of this link. In the theorem regarding the same, ...
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1answer
156 views

Pointwise estimate for a sequence of mollified functions

In the answer to Characterisation of one-dimensional Sobolev space Tomás wrote ... let $\eta_\delta$ be the standard mollifier sequence. Let $u_\delta=\eta_\delta\star u$ and note that for any ...
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1answer
108 views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
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0answers
23 views

Find $u:[0,T]\to H^2$ such that $u(0)=u_0\in H^2$ and $u_t(0)=u_1\in H^1$.

Let $u_0\in H^2$ and $u_1\in H^1$. If we define $$ \begin{align*}u:[0,T]&\longrightarrow L^2\\ t&\longmapsto u_0+\int_0^tu_1\;ds \end{align*}$$ then $u(0)=u_0$. Furthermore, the weak ...
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1answer
35 views

Strong maximum principle for weak solutions?

For a general linear parabolic equation, is a strong maximum principle possible when the solutions are merely weak solutions (i.e. they lie in a Bochner space)? Is there some proof possible that does ...
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0answers
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$\big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} \le C_{>0}\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$ holds? [duplicate]

I want to know that whether the following inequality holds or not for complex-valued functions $f_1$, $f_2$, $f_3$ on $\mathbb R$: $$ \big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\big\|_{H^1(\mathbb R)} ...
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1answer
102 views

Is a Sobolev function absolutely continuous with respect to a.e.segment of line?

Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes. It seems to me ...
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2answers
36 views

Continuation of $W^{1,p}_0$-functions

I know that for an open set $\Omega \subset \mathbb{R}^n$ and $1 \leq p \leq \infty$, a function $u \in W^{1,p}_0(\Omega)$ can be continued to a function $v \in W^{1,p}(\mathbb{R}^n)$ by setting $v=u$ ...
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About the denseness of $C^{\infty}_0 \cap W^{1,p}$ in $W^{1,p}$ [duplicate]

By the Theorem from Meyers-Serein, we know that for open $\Omega \subset \mathbb{R}^n$ and $1 \leq p < \infty$ the set $C^{\infty}(\Omega) \cap W^{1,p}(\Omega)$ is dense in the Sobolev space ...
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97 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
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0answers
27 views

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
29 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 ...
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2answers
35 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
59 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 compact support, dense in the set of smooth functions $C^\infty(\Omega)$ with respect to the norms $\|\cdot\|_{L^p}$ and the Sobolev-Norm ...
3
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2answers
145 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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1answer
77 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
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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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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 $u \in H^{s+l}$ ...
2
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1answer
41 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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36 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
38 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 ...
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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 ...
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1answer
36 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 ...
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33 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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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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56 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 ...
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48 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: ...
3
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1answer
44 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}, ...
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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33 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 ...
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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 ...