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.
1,032
questions
3
votes
1
answer
116
views
Does the union of fractional Sobolev spaces fills $L^p$?
Let $\Omega\subset \Bbb R^d$ be any open set. Recall that for $s\in (0,1)$, the fractional Sobolev space $W^{s,p}(\Omega)$ is the collection of function in $L^p(\Omega$ such that
\begin{align*}
\iint_{...
0
votes
1
answer
43
views
$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint.
Consider the problem
$$\Delta u = f \quad\text{in $\...
0
votes
1
answer
53
views
Uniform convergence of differential quotients in $L^1$
I know that the question arose already in other contexts. However, I think this question might be different. If I have $f\in W^{1,1}$ then it is obvious that $\frac{f(x+t)-f(x)}{t}$ converges ...
1
vote
0
answers
44
views
Extension operator maps which fractional Sobolev space to $W^{p,s}(R)$
Let us assume we have the following extension operator:
$$
\operatorname{ext}_R^\sigma u=
\begin{cases}
u(x) & \text{if }x \in (0,T)\\
u(0) & \text{if }x \in(0,T)^c.
\end{cases}
$$ We ...
2
votes
2
answers
192
views
$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?
Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has
$$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
2
votes
2
answers
179
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
2
votes
0
answers
50
views
Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...
0
votes
0
answers
208
views
Gauss transformation in fractional Sobolev space
Let $g_{\mu}(x) = \mu^{d/2}\exp(-\pi\mu|x|^2)$ for every $\mu > 0$. Prove that
$$
\int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}} u\right|^{2} \geq \int_{\mathbb R^{d}}\left|(-\Delta)^{\frac{s}{2}...
1
vote
0
answers
107
views
Is it possible to bound the L2 norm of the gradient of a divergent by the L2 norm of the Lapacian?
Is it possible to show for $u:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$
that
$$\||\nabla(\nabla\cdot u)|\|_2^2\leq C\||\Delta u|\|_2^2?$$
Here $\||f|\|_2$ is the norm in $(L^2(\Omega))^3$ and ...
2
votes
0
answers
69
views
Any solution of an evolution problem tends to a steady state in $L^2$?
I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
0
votes
0
answers
35
views
Finding an element of Gelfand triple with a designated time derivative
Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
where $V'$ is the dual of $V$ and the inclusions are ...
9
votes
1
answer
595
views
Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
3
votes
0
answers
73
views
Absolute continuity of $t \to \lVert u(t) \rVert^2_{H}$ and Gelfand triple : are they equivalent?
Let $V$ be a separable Banach space and $H$ be a separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
and the inclusion maps are continuous with dense images. Here $...
3
votes
1
answer
183
views
$L^{\infty}$ estimate for heat equation with $L^2$ initial data
Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem:
$$\begin{...
2
votes
0
answers
152
views
Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?
Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
1
vote
1
answer
113
views
Can functions with "big" discontinuities be in $H^1$?
How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
4
votes
1
answer
125
views
Embeddings of the maximal domain for the Laplacian
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:
$$D = \left\{ f \in L^2(\...
3
votes
2
answers
231
views
Can every $L^p$ function be written as the weak derivative of a Sobolev function?
Let $\mathbb B^n$ be the open unit ball in $\mathbb R^n$, and $g: \mathbb B^n \to \mathbb R^n$ a measurable function with $|g| \in L^p (\mathbb B^n)$. Does there exist some function $f$ in the Sobolev ...
1
vote
0
answers
70
views
On Talenti's proof of optimal constant in Sobolev inequality
I'm reading the paper by Giorgio Talenti on the best constant for the Sobolev inequality.
The main theorem states that for $u:\mathbb{R}^m\rightarrow \mathbb{R}$ sufficiently smooth (eg. Lipschitz) ...
3
votes
1
answer
198
views
Weighted Lebesgue space with exponential weights: smoothing effect and properties
I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...
0
votes
0
answers
45
views
Constants in the entropy number of the Sobolev space
For a Sobolev space with $W^s(\Omega)$, where $\Omega\subset R^d$ is a compact space with smooth boundary, we know that the entropy number satisfies $e(\delta, W^s(\Omega, 1),\|\cdot\|_{L_\infty})\leq ...
1
vote
0
answers
71
views
$T$ trace, then $Tg(u)=g(T(u))$ for all $u$ on $W^{1,p}$
The trace operator $T$ is defined for bounded domain $U$ with $C^1$ boundaries as the linear, continuous operator
$T: W^{1,p}(U) \rightarrow L^p(\partial U)$
such that
$$
Tu=u\;\text{ on }\partial U
$$...
2
votes
1
answer
90
views
Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function
Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
5
votes
0
answers
105
views
A function whose derivatives belong to $BMO(\mathbb{R}^n)$
I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert
s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp.
539-558. In this paper he defines ...
2
votes
0
answers
73
views
How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$
Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
0
votes
0
answers
45
views
Fractional Bochner spaces $H^s((0,\infty); V)$
I'm looking for "intrinsic" definitions of fractional Bochner spaces such as $H^s((0,\infty); V)$ for a Banach space $V$ and exponent $s \in (0,1)$. I have seen these spaces being defined as ...
2
votes
0
answers
61
views
Derivative of a functional involving integral and level set
Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional
$$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$
where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
2
votes
0
answers
139
views
$H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)
This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao.
There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
0
votes
0
answers
45
views
Weighted version of the Gagliardo Nirenberg inequality
I'm searching for a weighted Gagliardo-Nirenberg inequality similar as in https://arxiv.org/pdf/1307.1363.pdf where the weight is a power of the last component. Is there an inequality of the form
$$
\...
4
votes
1
answer
235
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
4
votes
1
answer
491
views
$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
In a paper I am writing I need to show that a certain real-valued function $f\in C^0(B_1)$ belongs to the Sobolev space $W^{1,2}(B_1)$ ($B_1$ is the unit ball). So far I have been able to show that ...
1
vote
0
answers
179
views
Is this a well known space? Perhaps homogeneous Sobolev-like space?
The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...
8
votes
1
answer
485
views
A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
1
vote
0
answers
85
views
Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
2
votes
1
answer
108
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
0
votes
0
answers
59
views
Friedrich's second inequality for functions with zero average
Friedrich's second inequality (or Maxwell Estimates or Gaffney’s inequality in the literature) is referred as follows: for all $\mathbf{u} \in H^1(\Omega)^2$ satisfying either $\mathbf{n} \cdot \...
2
votes
0
answers
165
views
finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
2
votes
0
answers
40
views
Blow up for certain classes of distributions
Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
2
votes
1
answer
148
views
A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
5
votes
1
answer
235
views
Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?
Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by
$$M f_{\varepsilon} (x) = \sup_{r \...
2
votes
1
answer
351
views
On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$
Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$.
I have found in several research ...
2
votes
0
answers
62
views
Localized estimate for divergence free vector field
Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
4
votes
1
answer
339
views
Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$
Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
0
votes
0
answers
117
views
About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"
I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1.
We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
1
vote
1
answer
113
views
Sobolev-type estimate for irrational winding on a torus
Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
2
votes
0
answers
128
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
3
votes
1
answer
188
views
Blowup of Sobolev norms in approximating a non-absolutely continuous function
Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that if $f$ fails to be absolutely ...
3
votes
1
answer
494
views
Existence of a curve of finite length on the image of an embedding which is Sobolev
Suppose that we have an embedding $f:\mathbb{R}^2\to\mathbb{R}^3$ which belongs in the Sobolev space $W^{1,p}_{loc}(\mathbb{R^2},\mathbb{R}^3)$ for some $p>2$. Is it true then that for any two ...
5
votes
0
answers
197
views
Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
2
votes
0
answers
70
views
Sobolev inequalities in weighted Sobolev spaces
My definition for "weighted Sobolev space" $W^{1,p} _w (\Omega)$ is just a set of functions with the property that they admit distributional derivatives and that
$$ \int_\Omega |f|^p (x) w(x)...