All Questions
Tagged with fa.functional-analysis sobolev-spaces
278 questions with no upvoted or accepted answers
2
votes
0
answers
229
views
Weighted Sobolev norm in terms of Spherical harmonics coefficients
Let $M = [1,\infty) \times S^2$.
Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...
- 969
2
votes
0
answers
225
views
Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$
Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and
$$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$
for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
- 233
2
votes
0
answers
65
views
Regularity of solution to first order time dependent variational problem
Consider the following first order evolution problem over some regular bounded domain $\Omega\subset\Bbb R^d$
$$\frac{\partial\phi}{\partial t}(\mathbf{x},t) +\vec V(\mathbf{x},t)\cdot \nabla\phi(\...
- 153
2
votes
0
answers
94
views
From some priori estimates can we estimate higher Sobolev norm?
Suppose $u$ is a smooth function on bounded set $\Omega$ with smooth boundary such that
$$\|u\|_{W^{1,p}(\Omega)}\le C\|\phi\|_{W^{1-1/p,p}(\partial\Omega)}$$
where $u|_{\partial\Omega}=\phi$.
Can we ...
- 309
2
votes
0
answers
125
views
Regularity up to boundary of a solution $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ to $\Delta^2 u = -\text{div}\, F$
Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system
...
- 1,245
2
votes
0
answers
57
views
Is this Beppo-Levi curl space a Banach space?
Let us define the quotient space:
$$ 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
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
$$
\|fg\|_{L^{q,r_3}(\mathbb{R}^n)}\lesssim \|f\|_{B^{n/p}...
- 282
2
votes
0
answers
223
views
Interpolation of embedded Hilbert spaces and intersection
I'm wondering under what hypothesis it is true a property like
$$[\mathcal{H}_1\cap X, \mathcal{H}_2\cap X]_{\theta}=\mathcal{H}_1\cap X\cap [\mathcal{H}_1, \mathcal{H_2}]_{\theta}$$
where $\mathcal{H}...
- 81
2
votes
0
answers
1k
views
Compact embedding of the Sobolev space $H^m(\Omega)$ and $L^2(\Omega)$ from Rellich-Kondrachov theorem
From the Rellich-Kondrachov theorem we know that $H^m(\Omega)\hookrightarrow_c L^2(\Omega)$ when $\Omega$ is bounded of class $C^1$ and $m\geq 1$ is an integer. Also this is not true if $\Omega:=\...
- 657
2
votes
0
answers
42
views
Generalized Hardy operator and Lorentz gamma spaces
I would like to find an inequality which would 'place' the generalized Hardy operator $\int_0^th(y)dy\int_y^tk^*(s)ds$ in between two Lorentz gamma spaces.
Any literature or ideas would be greatly ...
- 109
2
votes
0
answers
395
views
A question in Sobolev spaces involving time
Let $X$ be a Banach space, we understand $L^1(0, T, X)$ is the space of strongly measurable functions from $[0, T]$ valued in $X$, that is integrable. Assume ${\bf u}\in L^1(0, T, X)$, we say ${\bf v}\...
- 637
2
votes
0
answers
137
views
Conditions on the inequality with a gauge norm
Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm
$$
\rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\...
- 343
2
votes
0
answers
160
views
Approximation in fractional Sobolev space
Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$.
How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(\Omega)}= ...
- 1,101
2
votes
0
answers
159
views
Explicit homeomorphism between $L^p$ and Sobolev Space
From the Anderson-Kadec theorem, we know that all separable infinite-dimensional Banach spaces are homeomorphic. I'm wondering, is there an explicit such homeomorphism between $W^{p,k}(\mathbb{R}^n)$ ...
- 5,405
2
votes
0
answers
115
views
If $u_n \to u$ in $H^1_0(\Omega)$, does $\chi_{\{u_n = 0\}} \to g$ for some $g$ in some space, for a subsequence?
Let $\Omega$ be a bounded and smooth domain.
Suppose we have $u_n \to u$ in $H^1_0(\Omega)$. We know that for a subsequence, $\chi_{\{u_n = 0\}} \rightharpoonup f$ to some $f$, weak-* in $L^\infty(\...
2
votes
0
answers
90
views
Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces
In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
- 5,405
2
votes
0
answers
274
views
Gagliardo-Nirenberg inequality for periodic functions?
I am interested in a Gagliardo-Nirenberg type inequality for functions in the space
$$H^1_T(\mathbb{R}^n)=\lbrace \psi=u+iv\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)\rbrace$$
...
- 383
2
votes
0
answers
240
views
Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
2
votes
0
answers
279
views
Relationship between $p$-capacity and Riesz $s$-capacity of a set
What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms ...
- 839
2
votes
0
answers
54
views
example in $L^p_{s}-$Sobolev spaces
We define $L^p-$
Sobolev spaces as follows:
$$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$
where $\langle \...
- 305
2
votes
0
answers
255
views
Sobolev Multiplication on non-compact manifold
We know that for a compact Riemannian $n$-dim manifold $(M,g)$(the boundary could be nonempty), the Sobolev Multiplication Theorem states that $L^p_k\times L^q_l⟶L^r_m$, where $1/r−m/n>1/p−k/m+1/...
- 1,915
2
votes
0
answers
81
views
Sobolev extension with boundary condition
Let $\Omega$ be a Lipschitz bounded domain of $\mathbb{R}^n$, divided in two Lipschitz subdomains $\Omega_1$ and $\Omega_2$ such that $\Omega_1 \cap \Omega_2 = \emptyset$. We define the following ...
- 143
2
votes
0
answers
331
views
Sobolev embeddings for vector-valued functions
I would like to know if there is a simple extension of the standard Sobolev embeddings for functions taking values in another Euclidean space.
In particular, let $\Omega \subset \mathbb{R}^n$ be a ...
2
votes
0
answers
672
views
Intersection of Sobolev space with the space of continuous functions
While doing some problems, I came across the space $H=H^1(\Omega) \cap C(\Omega)$, where $\Omega$ is subset of $\mathbb{R^n}$. So far, by definition of these subspaces, We know that none of these are ...
- 141
2
votes
0
answers
116
views
Density of $C^0(\Bbb R^{n}\times (0,T))$ and $C^{\infty}_c(\Bbb R^{n}\times (0,T))$ in $L_{p,q}(\Bbb R^n_T)$
Let the space $L_{p,q}(\Bbb R^n_T)$ be defined as the set of all measurable $f:\Bbb R^{n}\times(0,T)\to\Bbb R$ such that $||f||_{p,q}<\infty$, where
$$
||f||_{p,q}:=\left(\int_0^T\left( \int_{\Bbb ...
- 1,245
2
votes
0
answers
65
views
Generalized Besov spaces with different integrability and smoothness in space and time?
Consider the family of Besov spaces $B_{p,q}^{s}(\mathbb{R})$ with $0<p,q \leq \infty$ and $s \in \mathbb{R}$.
Is there a natural way to define spaces of generalized functions $f(t,x) \in \mathcal{...
- 2,306
2
votes
0
answers
2k
views
Reference for a proof of the Gagliardo-Nirenberg Interpolation Inequality?
In the book Linear and Quasi-linear Evolution Equations in Hilbert Spaces by Cherrier and Milani, Theorem 1.5.2, we are given the following version of the GN interpolation inequality:
Let $\Omega\...
- 1,247
2
votes
0
answers
194
views
A question regarding mollifiers on Sobolev spaces on closed manifolds
Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
- 505
2
votes
0
answers
115
views
Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
- 305
2
votes
0
answers
64
views
The continuity of $L^2$ gradient on moving domain
I post this on MSE too. Sorry for this short cross posting. But I stuck in this problem for a while... I don't know what the right tool should be used to handle the moving domain problem...
Let $I:=(...
- 679
2
votes
0
answers
207
views
Smoothing properties of analytic semigroups
Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates
$\|T_p(t)f\|_{...
- 21
2
votes
0
answers
166
views
Getting an a priori energy estimate from PDE weak formulation
On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying
$$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv$...
2
votes
0
answers
82
views
Properties of a Sobolev bound
I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...
- 193
2
votes
0
answers
184
views
Modify the jump set of $BV$ function
Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...
- 679
2
votes
0
answers
552
views
Normed space between $H^{0+}$ and $L^2$
Cosider a function $f\in L^2(\mathbb{R}^3)$ with consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty \, .\qquad\mbox{(*)} \, $$
Of course if $f\in ...
- 943
2
votes
0
answers
93
views
Multiplier operators on anisotropic weighted $L^2$ spaces
Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$,
$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$
where the scalar complex function ...
- 21
2
votes
0
answers
150
views
Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
- 371
2
votes
0
answers
491
views
$L^\infty-L^1$ smoothing effect for the heat equation
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation
$$u_t - \Delta u = 0$$
$$u(0) = u_0$$
where $u_0$ is bounded initial data. Here ...
- 21
2
votes
0
answers
69
views
Doubts regarding pre-compactness of bounded sequence of measure valued functions
Given:
$\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ .
...
- 121
2
votes
0
answers
114
views
What is $(L^2(M), H^1_0(M))_{\frac 12}$ on a smooth manifold with boundary?
Let $M$ be a smooth compact manifold. If $M$ is closed, we have that the interpolation space
$$(L^2(M), H^1(M))_{\frac 12}=H^{\frac 12}(M)$$
(see Taylor's book on PDE for example). Suppose $M$ has a ...
- 251
2
votes
0
answers
142
views
Points are removable for weakly differentiable functions
If $\Omega \subseteq \mathbb{R}^N$ is an open set and $N \ge 2$, then any point $a \in \Omega$ is removable for weakly differentiable maps: for each function $u \in W^{1, 1} (\Omega \setminus \{a\})$, ...
- 2,834
2
votes
0
answers
223
views
One parameter family of elliptic equations
Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = N(\...
- 21
2
votes
0
answers
467
views
Reference request: The compactness and compact embedding in Besov Space?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...
- 679
2
votes
0
answers
396
views
"Friedrichs extension Laplacian" vs "Weak Laplacian" and fractional powers
Take $\Omega$ to be a bounded domain and consider Neumann BCs.
In some works, I see that a Laplacian $(-\Delta_D)^{\frac 12}$ is defined as an operator with domain $H^1(\Omega)$, and in other works, ...
- 21
2
votes
0
answers
52
views
About norm on $H^{\frac 12}(M \times \{0,1\})$
Let $X=M \times \{0,1\}$ with $M$ a smooth compact manifold without boundary.
Define the fractional Sobolev space $H^{\frac 12}(X) = (L^2(X), H^1(X))_{\frac 12}$, as the real interpolation space ...
- 171
2
votes
0
answers
320
views
Sobolev trace theorem
Set $Q:=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$,
where $\Omega$ is knows as a bounded domain
with smooth boundary $\partial D$.
We choose any subdomain $D\subset Q$
with smooth boundary $\partial ...
- 375
2
votes
0
answers
102
views
Sobolev trace of $H^1(\mathcal{M} \times I)$ functions
Let $\mathcal{M}$ be a compact Riemannian manifold and let $I=(0,1)$. I seek a trace theorem saying that functions $u \in H^1(\mathcal{M} \times I)$ have a well-defined trace at $\mathcal{M} \times \{...
- 21
2
votes
0
answers
119
views
Find $U \in H^1(\Omega \times (0,\infty))$ such that $\nabla E(u-\bar u)\nabla U \geq 0?$ (PDE harmonic extension)
Let $\Omega$ be a bounded smooth domain. Given $u \in H^{\frac 12}(\Omega)$ with mean value $\bar u = 0$, let $Eu = v \in H^1(\Omega \times (0,\infty))$ solve
$$\int_0^\infty\int_\Omega \nabla v\nabla ...
- 49
2
votes
0
answers
553
views
Sobolev space for manifold with boundary
For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...
- 1,069
2
votes
0
answers
382
views
Sobolev space and trace theorems on a non-compact Riemannian manifold with boundary ($M \times (0,\infty)$)
Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$.
I am ...
- 53