# Questions tagged [sobolev-spaces]

A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.

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### Convolution with an analytic semigroup

Let $e^{At}$ denote an analytic semigroup on Hilbert space $X$ generate by $A:D(A)\to X$. Also, let $f\in L^1(0,\tau;X)$. I want to show that the convolution
$$ g(t)=\int_0^t e^{A(t-s)}f(s)ds$$
...

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### An open problem in Sobolev spaces

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator
$$
E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n)
\quad
\text{and}
\quad
E:W^{1,q}(\Omega)\to ...

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### Extended Global approximation theorem

In Evans,
$\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...

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### Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder

I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions
$$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$
$$\int_{\mathbb{R}^+} f \in \...

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### The space $H_\Delta(\Omega)$

I'm looking for good references for the study of the following space
$$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$
especially the trace theorems with proofs, where $\Omega \...

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### Estimation fractional Sobolev norm by Littlewood-Paley projection

Let $1 < p < \infty$, $s > 0$, $f\in S'(\mathbb{R}^d)$ with sobolev norm
$\|f\|_{W^{s, p}} = \|(1-\Delta)^{s/2}f\|_p = \|(1 + 4\pi^2 |\xi|^2)^{s/2} \hat{f}(\xi))^\check{} \|_p$.
We need to ...

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109 views

### Existence and regularity for fractional Poisson-type equation

According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results.
Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...

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### Specific Bounds on Divergence operator in Sobolev setting

Given that $\vec{u}=(u_1,u_2)\in H^1(\Omega)\times H^1(\Omega)$ where $\Omega\subseteq\mathbb{R}^n$ is open and bounded with Lipschitz boundary, does there exist specific inequalities which links $\...

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### Regularity in Orlicz spaces for the Poisson equation

I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007)
$$\Delta u=f \quad \quad \quad \quad (1)$$
Lemma 2: There is a constant $N_1 >1$ so that ...

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### Interpolation inequalities involving mean curvature operator

Are there any interpolation inequalities (for example, of Gagliardo-Nirenberg type) involving the mean curvature operator
$$\mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right)$$
(in any ...

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### Unboundedness of the Sobolev norm of a sequence : Does it follow from Sobolev embedding?

Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ ...

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### A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?

Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...

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### Density on a specific functional space.

I have a question about density. It's probably trivial but I am just learning functional analysis so nothing is trivial to me. Here is my question. Let
$$
\mathcal{X}\colon=\mathcal{H}^1(0,1;\mathbb{...

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### Making the Fourier transform quantitative

I am undergraduate Physics student and understand that this is a professional mathematics forum. But due to perhaps broader interest, I hope this question is suitable for this website.
I understand ...

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### Fractional embedding inequality with $L^{\infty}$ norm

Here we consider the fractional Sobolev spaces and suppose $u$ is a vector function in $\mathbb R^2$. For $q>2$, is the following always true? $$\Vert Du \Vert_{L^{\infty}(\mathbb R^2)} \leq C\Vert ...

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### Bounded deformation vs bounded variation

Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...

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### Removable set for Sobolev space

It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{...

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### $f_n$ is bounded in $C(0,T;H^2(0,L))$ so is $f_n^p$?

Let $1<p<\infty$, and $f_n$ be a bounded sequence in $C(0,T;H^2(0,L))$. It looks obvious to me that $f_n^p$ is also bounded in $C(0,T;H^2(0,L))$. When we take the derivative of $f^p(t)$ twice we ...

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### Positive splitting of Sobolev convergence

Let $f,g,h \in H^1(\mathbb{R}^n)$ be non-negative Sobolev functions wuch that $f^2 = g^2 + h^2$. Let also $\{f_k\} \subseteq H^1(\mathbb{R}^n)$ be non-negative Sobolev functions such that $f_k \to f$ ...

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### Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$
is not ...

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### Superposition operator from Sobolev space to Lebesgue space

Let $\Omega$ be a bounded, connected set in $\mathbb{R}^2$ with smooth boundary. I am wondering under what conditions on the real function $f(x):\mathbb{R}\to \mathbb{R}$ the superposition operator $F(...

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### Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...

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### Why is $H^{1/2}$ a Hilbert space?

Let $n\in\mathbb{N}$ and $\Omega \subseteq \mathbb{R}^n$ sufficiently smooth. Then we have the Hilbert space $H^1(\Omega)$ and the trace operator $\operatorname{tr}: H^1(\Omega) \to L^2(\partial \...

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### Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$

Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $...

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### How do we know the mollification is in the Sobolev space?

The first theorem of section 5.3. in Evan's PDE discusses approximating a function in $W^{p, k}$ by it's mollifications.
Suppose $k$ is a positive integer, $1\leq p <\infty$ and $U$ is an open ...

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### Sobolev extension with boundary condition

Let $\Omega$ be a Lipschitz bounded domain of $\mathbb{R}^n$, divided in two Lipschitz subdomains $\Omega_1$ and $\Omega_2$ such that $\Omega_1 \cap \Omega_2 = \emptyset$. We define the following ...

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### Measurability of specific function

Let $I\subset\mathbb{R}$ denote an open and bounded interval of the real line, $H_0^1(I)$ all quadratic integrable Sobolev functions and $C(\bar{I})$ all continuous functions on said interval.
Since ...

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### What is a relation between energy space and $L^p_s-$Sobolev spaces?

We define energy space
$$E= \left\{ f\in \mathcal{S}'(\mathbb R^d): \|\nabla f\|_{L^2} + \|xf\|_{L^2} < \infty \right\}.$$
and
Sobolev spaces
$$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^...

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### A question on existence of a Sobolev Hilbert space, where convergence implies uniform convergence [closed]

Is there a Sobolev Hilbert space $H^k(\Omega)$($\Omega$ open subset of $\mathbb{R}^m$, with a smooth boundary), for some $k \in \mathbb{N}$, such that, any sequence in the space $C^0(\bar{\Omega})\cap ...

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### inverse of sobolev riemannian metric still sobolev?

Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth ...

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### Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...

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### Approximation in parabolic Sobolev spaces

I am given the following function: fix any $p$ and $\beta \in (0,1)$
$$
u \in L^{p-\beta}(0,T;W^{1,p-\beta}(\Omega)) \quad \text{and} \quad \frac{du}{dt} \in L^{\frac{p-\beta}{p-1}}(0,T; W^{-1,\frac{...

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### Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.
Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...

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### Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...

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### Is a relatively weakly compact subset of $W^{1,1}(\Omega)$ metrizable?

Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set.
Is it true that $(S,w)$ is metrizable?
Since $S$ is relatively weakly compact, it ...

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### Steklov averages and negative parabolic sobolev spaces

Suppose one is given a function
$$
w \in L^p(0,T;W^{1,p}(\Omega)) \qquad \text{and} \qquad \frac{dw}{dt} \in L^{p'}(0,T; W^{-1,p'}(\Omega))
$$
I am interested if the following holds:
Denote the ...

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### Sobolev embeddings for vector-valued functions

I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space.
In particular, let $\Omega \subset \mathbb{R}^n$ be a ...

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### PDE satisfied by projection of a function onto a subspace

Given an open bounded set $D\subset \mathbb R^N$, let $f\in W^{-1,q}(D)$ and let $u$ be a Sobolev function $u\in W_0^{1,p}(D)$ such that $u$ solves the PDE
$$
\begin{cases}
-\Delta_p u=f\;\text{in $D$}...

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### Approximating norms using numerical integration? [closed]

I have a sequence of functions $u_m$ in $H^1(\Omega)$, where $\Omega$ is Lipschitz such that $u_m(x)=\int_{|y|\le \epsilon} \, f_m(x,y) \, dy$, but the integral cannot be expressed in terms of ...

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### Fast growing unbounded functions in the Sobolev space $H^1(\Omega)$

I am looking for unbounded functions that grow rapidly fast near the origin, but are in the Sobolev space $H^1{(\Omega)}$, where $\Omega$ is a unit square centered at the origin.
I already know about ...

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### Estimates for the Sobolev inequality

How to prove the Sobolev estimate:
If $\Omega$ is a bounded open subset of $\mathbb R^N$, then
for any $q>1$
$$
\|u\|_{L^{q}(\Omega)} \leq C|\Omega|^{1/q} q^{1- 1/N}\| \nabla u \|_{L^{N} (\Omega)...

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### Weak closedness of the extremal set to a linear inequality

Let $\mathcal{H}$ be a Hilbert space, let $p\ge 2$, and consider a bounded linear operator
$
T\colon \mathcal{H}\to L^p(\mathbb R^d). $
Is the set $M=\{f\in \mathcal H\ :\ \|Tf\|_{L^p}= \|T\|\|f\|...

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### Intersection of Sobolev space with the space of continuous functions

While doing some problems, I came across the space $H=H^1(\Omega) \cap C(\Omega)$, where $\Omega$ is subset of $\mathbb{R^n}$. So far, by definition of these subspaces, We know that none of these are ...

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### Smallness of cut-off functions at critical Sobolev regularity

Consider the class of functions
$$X:=\{f\in \mathcal{C}_0^{\infty}(\mathbb{R})\;s.t.\;f\equiv 1 \mbox{ in a neighbourhood of}\;\;x=0\}$$
Is it true that, for every $\varepsilon > 0$, I can find $...

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### Regularity of the Jacobian of a $W^{2,n}$ Sobolev mapping

Given a mapping in the Sobolev space $f\in W^{2,n}_{\rm loc}(\mathbb{R}^n,\mathbb{R}^n)$ I would like to know what is the
Sobolev regularity of the Jacobian $J_f=\operatorname{det} Df$.
It is well ...

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### A Poincaré-type inequality: proof or counterexample

The following is a simplified version of a Poincaré-type inequality that I'm studying; I'd like to prove it (the inequality) or find a counter example. Consider a function $f:[0,1]^2\rightarrow\mathbb{...

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### Existence for $-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$

Let $\Omega$ be a smooth bounded domain. Consider the equation
$$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$$
$$u|_{\partial\Omega} = 0$$
where $f,g$ are smooth functions on $\Omega$ and $\varphi$...

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### Is a Sobolev map with smooth minors smooth on the whole domain?

This question is related (but not identical) to this question.
Let $d>2$ be an integer and let $2 \le k \le d-1$ be a fixed integer. Suppose that at least one of $k,d$ is not even. Let $\Omega$ be ...

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### Fourier Lapalacian over periodic end

This is a technical question on Taubes' paper: Gauge theory over periodic end. on Page 378.
Recall that:
Let $Y$ be a closed manifold, with $b_1=1$, and $\tilde Y$ be the $\mathbb Z$-covering of $...

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### Relation between a norm and norm of Besov spaces

Let $(H, \|\cdot\|)$ be a Hilbert space, $A \colon D(A)\subset H \longrightarrow H$ generates an analytic semigroup $T(t)$ on $H$. We define the following Banach space with the respect norm
$$F=\{x\in ...