All Questions
Tagged with fa.functional-analysis sobolev-spaces
652 questions
0
votes
1
answer
96
views
Interpolated Sobolev norm inequality
Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$...
- 13
3
votes
0
answers
181
views
Variational problems living in two different Sobolev spaces
Is there a general reference concerning variational problems living in $W^{h,p}\times W^{k,p}$, with $h, k\in\mathbb{N}_0$ not coinciding? I'm thinking to problems of type:
$$\inf_{u,v}\int_{\Omega} ...
- 4,459
6
votes
0
answers
113
views
A continuity argument for a dispersive $gKdV$ estimate
I'm learning about the gKdV equation, following Schlag & Muscalu vol II. We're looking at
$$\begin{cases} u_t + u_{xxx} + F(u)_x = 0 \\ u_0 = g\end{cases}$$
where $F(u) = u^5$ (for example). The ...
- 163
2
votes
1
answer
174
views
Is the graph of a Sobolev function path connected?
Let $\Omega$ be a bounded, open, simply connected subset of $\mathbb R^n$ with Lipschitz boundary.
Question: Does every function in the Sobolev space $W^{1,1} (\Omega)$
admit a representative whose ...
- 6,155
4
votes
0
answers
310
views
Sobolev spaces and spectral theorem
Consider a generalised harmonic oscillator of the form
$$
A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n,
$$
where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The ...
0
votes
0
answers
247
views
Imbed Sobolev spaces of fractional order into Holder spaces?
This result exist (https://encyclopediaofmath.org/wiki/Imbedding_theorems ) for regular (i.e. not fractional) Sobolev spaces; looks like it's provable for fractional spaces through results for Besov ...
- 93
1
vote
0
answers
56
views
Monotonically increasing and bounded function is in $BV_{loc}$?
For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$ be monotonically increasing and $\lim_{x\to -\infty} f_n(x)=0$ and $\lim_{x\to \infty} f_n(x)=1$. It follows $f_n$ is differentiable a.e..
I'm ...
- 173
2
votes
1
answer
697
views
Confusing definition of homogeneous Sobolev norm of order -1
Let $\Omega \subset \mathbb{R}^{d}$ and $\|.\|$ is the standard euclidean $2$-norm. I came across a definition of $\dot{H}^{-1}(\Omega)$ which is a bit confusing. In [1] authors define the following ...
- 407
0
votes
0
answers
67
views
Multiplication of a Riesz basis
Let ${(\phi_n(.),\psi_n(.))}_{n\geq 1}$ be a Riesz basis in $H^1_0(0,1) \times L^2(0,1)$.
My question is the following: If we multiply the basis by the matrix $e^{Mx}$, $x \in (0,1)$ where $M$ is a ...
- 617
-1
votes
1
answer
118
views
Sobolev injections [closed]
It is true to write that
$W^{1,\infty}(]0,\infty[) \hookrightarrow C([0,\infty[)$ et $W^{1,1}(]0,\infty[) \hookrightarrow C([0,\infty[)$ ?
Thanks
- 19
1
vote
0
answers
223
views
Is this mixed-Sobolev interpolation inequality valid?
By $L^p_T\dot H^s$ I mean the space $L^p([0, T] \to \dot H^s)$ where as usual
$$\|u\|_{\dot H^s}^2 = \int_{\mathbb R^d} |\hat u(\xi)|^2 |\xi|^{2s} ~d\xi.$$
Of course, we can interpolate in the time ...
- 708
1
vote
1
answer
398
views
Convolution mollification of $H^s$ functions uniformly in the unit ball of this sobolev space
Let $\phi$ be a nonnegative $C_c^\infty(B(0,1))$ function, where $B(0,1)\in \mathbb R^n$ is the unit ball, and $\int \phi =1$. Let $\phi_{\epsilon}(x) =\epsilon^{-n} \phi(x/\epsilon).$ For any $L^2$ ...
- 1,755
1
vote
0
answers
89
views
Derivation in Sobolev space [closed]
Let $f\in W^{1,\infty}(]0,T[)$ $(0<T\le\infty)$ such that
$f(x)>0$ a.e. $x\in\mathopen]0,T[$ and let
$$g(x)=e^{-\int_0^x \frac{ds}{f(s)}}$$
Formally $g' = -\frac{1}{f}g$.
How can I justify this ...
- 19
2
votes
0
answers
57
views
Is this Beppo-Levi curl space a Banach space?
Let us define the quotient space:
$$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
- 163
5
votes
1
answer
453
views
Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above ...
- 1,101
1
vote
0
answers
130
views
Fractional Sobolev embedding theorem
Let $\psi \in C^\infty_c(\mathbb R^N)$ be a test function with support iN $B(0,R)$. Is it true that the following inequality holds
$$\int_{B(0,R)} \psi^2 u^{\frac{4}{1+\beta}} dx \le R^{1+\beta} \int_{...
- 99
2
votes
0
answers
72
views
Product of Besov and Lorentz functions
Let us fix $n\in\mathbb{N}^+$ and $p,q\in [1,\infty)$. Given $r_1,r_2,r_3\in[1,\infty)$, I would like to understand whether we have the bound
$$
\|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
- 282
0
votes
1
answer
345
views
Embedding of fractional Sobolev space into BMO
Is it true that $$\Vert u \Vert_{BMO(\mathbb R^2)} \lesssim_{s} \Vert u \Vert_{\dot H^s(\mathbb R^2)},$$
for $s \in (0,1)$, where $\dot H^s(\mathbb R)$ is the homogeneous fractional Sobolev space?
- 99
4
votes
0
answers
143
views
Sobolev space of maps between manifolds with boundary
Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds with non-empty smooth boundary.
If we consider the Sobolev space $W^{1,p}(M,N)$, is there a reference
on how to model this as a manifold?
If ...
- 3,423
2
votes
1
answer
1k
views
Weak derivatives and Sobolev spaces on Riemannian manifolds
I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.
Let $(M,g)$ be smooth ...
- 319
3
votes
1
answer
1k
views
Friedrichs mollifiers and Sobolev spaces
$\renewcommand{\epsilon}{\varepsilon}$The following is from John Roe's book Elliptic operators, topology and asymptotic methods. $S$ is a vector bundle on a compact manifold $M$, but I think for my ...
- 1,141
0
votes
0
answers
299
views
Some density properties about Sobolev periodic spaces
Let $L>0$ fixed. Consider the space
$$
\mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}.
$$
For $r \in \mathbb{...
- 205
2
votes
0
answers
223
views
Interpolation of embedded Hilbert spaces and intersection
I'm wondering under what hypothesis it is true a property like
$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$
where $\mathcal{H}...
- 81
4
votes
0
answers
176
views
If $u \in W^{\alpha,p}(0,T;X)$ for $\alpha \in (0,1)$, then $f(u) \in W^{\alpha,q}(0,T;Y)$ for some good $f:X \to Y$
Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij ...
- 51
0
votes
1
answer
110
views
Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$
Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
- 131
2
votes
1
answer
404
views
Trace of a function
Let $T,L> 0$ two real numbers and we consider the Sobolev space $X := L^2(0,T; H^1(0,L))\cap H^{1}(0,T;H^{-1}(0,L))$. My question is:
Given $f \in X$, the trace $ t \mapsto f(t,L)$ belongs to what ...
- 29
1
vote
1
answer
78
views
Is there any quantitative relationship between the two terms of a Helmholtz decomposition?
Let $\Omega \subset \mathbb R^3$ denote an open, bounded and simply connected set with smooth boundary. The Helmholtz decomposition
$$ L^2(\Omega) = \nabla H^1_0(\Omega) \oplus L^2(\operatorname{div}=...
- 163
4
votes
1
answer
255
views
Regularity of Nemitskii maps on Sobolev spaces
Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$).
Let $X=W^{1,p}(\Omega)$ with $p>1$ be the ...
- 41
2
votes
0
answers
1k
views
Compact embedding of the Sobolev space $H^m(\Omega)$ and $L^2(\Omega)$ from Rellich-Kondrachov theorem
From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...
- 657
1
vote
0
answers
81
views
Compact imbedding for weight space
We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define
$$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \...
- 183
2
votes
1
answer
160
views
Minimum solution over closed ball of $H_0^1(\Omega)$
Since more than 4 months ago, I have posted a question on Mathstack and I haven't recieved any concrete answers. The link to the original post with the problem and my attempts are here.
To summarize, ...
- 123
3
votes
1
answer
251
views
Parabolic Sobolev inequality in Sobolev mixed norm spaces
Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and
$$
1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}.
$$
Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^...
- 515
1
vote
0
answers
119
views
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
$\DeclareMathOperator\rad{rad}$
Can $H_{\mathrm{rad}}^s(\mathbb{R}^n)$ be compactly embedded in $L^{\sigma}(\mathbb{R}^n)$?
In $H_{\rad}^1(\mathbb{R}^3)$, by Struass estimate $|f(x)| \lesssim |x|^{-1} ...
- 429
4
votes
1
answer
367
views
Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain
Let $s>0$, $1<p<\infty$ and let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. Set $H^{s,p}(\Omega)=\{u\in L^p(\Omega):\exists\tilde u\in L^p(\mathbb R^n),\tilde u|_{\Omega}=u,(I-\...
- 938
2
votes
1
answer
205
views
Estimates for an elliptic PDE
Let's say I have an equation of the form $\Delta A = J$ where $J=u\nabla u + A|u|^2$ (Clarification: We are on $\mathbb{R}^3$ and $u$ is assumed to be in $H^1(\mathbb{R}^3)$). Then I could simply ...
- 469
3
votes
1
answer
426
views
Explicit example $f_k \to f$ converging strongly in $L^6(R^3)$, but only weakly in $H^1(R^3)$
Can we find an explicit example of a sequence of functions $f_k \in H^1({\mathbf R}^3)$ such that, $f_k \rightharpoonup f$ weakly converges in $H^1({\mathbf R}^3)$ and $f_k \to f$ strongly converges ...
- 543
1
vote
0
answers
74
views
Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 339
1
vote
1
answer
277
views
Checking the uniform denseness of a set in $C([0, 1], \mathbb{R}^2)$
Let $\lambda:[0, 1]\to \mathbb{R}$, and $b_{1j}, b_{2j}:[0, 1] \to \mathbb{R}$, $j = 1, \ldots, m$ be smooth functions. Consider the following two sets
$$\begin{align*}
S_1 &= \left\{ \begin{...
- 119
2
votes
0
answers
42
views
Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
- 109
1
vote
0
answers
441
views
Stein's extension operator for fractional Sobolev spaces
In his book Singular Integrals and Differentiability Properties of Functions,
Stein constructs an extension operator $\mathcal{E}:W^{m,p}(\Omega)\rightarrow
W^{m,p}(\mathbb{R}^{N})$, $m\in\mathbb{N}$, ...
- 411
6
votes
1
answer
378
views
Optimal constant in Sobolev embedding
It is well-known that the Sobolev space $H^1(0,s)$ embeds continuously in the space of continuous functions $C[0,s]$; in fact, Marti has found in 1983 that the optimal embedding constant is $\sqrt{\...
- 3,388
2
votes
1
answer
168
views
A question about series involving a Sobolev functions
Let $\Omega\subset\mathbb{R}^n$ open, bounded and smooth. Let $\lambda_j$ and $e_j$, $j\in\mathbb{N}$, be the eigenvalue and the corresponding eigenfunctions of the Laplacian operator $-\Delta$ in $\...
- 339
1
vote
0
answers
91
views
A bilinear estimates involving critical Sobolev norms
Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering ...
- 943
1
vote
0
answers
202
views
Uniformly local Sobolev spaces and interpolation
Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space
$$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|...
- 943
10
votes
1
answer
486
views
Sobolev inequalities on manifolds: dependence of the constants on the Riemannian metric
Let $g$ be a smooth Riemannian metric on the 2-torus $T^2$. $g$ induces the Sobolev space $W^{2,2}_g(T^2)$ via the norm
$$
\|f\|_{W^{2,2}_g}^2 = \int_M |f|^2 + g(\nabla^2 f,\nabla^2 f)\, \text{vol}_g,
...
- 381
2
votes
0
answers
395
views
A question in Sobolev spaces involving time
Let $X$ be a Banach space, we understand $L^1(0, T, X)$ is the space of strongly measurable functions from $[0, T]$ valued in $X$, that is integrable. Assume ${\bf u}\in L^1(0, T, X)$, we say ${\bf v}\...
- 637
3
votes
0
answers
217
views
Hardy Littlewood maximal function bounds
Let $u \in W^{1,p}(\mathbb{R}^n) \cap L^{\infty}(\mathbb{R}^n)$ be a given function for some $1<p< \infty$ and let $k \in \mathbb{R}$ be any number and consider the following maximal function
$$
...
- 455
7
votes
2
answers
3k
views
Arzelà-Ascoli theorem and Hölder spaces
Let $B\subset \mathbb{R}^n$ be a open ball. Let $\{f_i\}$ be a sequence of functions bounded in the Hölder norm $C^{k,\alpha}(B)$ for a given integer $k\geq 0$ and $\alpha\in (0,1)$.
Does there exist ...
- 21.8k
1
vote
1
answer
184
views
Example when Kantorovich condition would not hold
Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator
$$
(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.
$$
Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
- 343
2
votes
0
answers
137
views
Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
- 343