All Questions
Tagged with fa.functional-analysis sobolev-spaces
652 questions
1
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1
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184
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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
1
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1
answer
84
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A completion equality in product of Sobolev spaces
Let $\Omega$ be a nice bounded domain of $\mathbb{R}^n$. For $s\ge 0$ we define
$$H_{0}^{s}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}^{H^{s}(\Omega)}.$$
My question: is the following equality true?
$$...
- 81
1
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1
answer
219
views
Harmonic functions vanishing on the boundary and distance function asymptotics
Let $\Omega \subset \mathbb R^N$ be a $C^2$ domain. Let $u$ be a function such that $u \in W^{2,2}(\Omega)$ and $u = \Delta u = 0$ on $\partial \Omega$. Is it true that $$ c \le \frac{u}{[\mathrm{dist}...
user139845
1
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1
answer
125
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Sobolev extension operators
Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary and that $M\subset \tilde{M}$ with $(\tilde{M},g)$ also a compact Riemannian manifold with smooth boundary.
Let us consider a one-...
- 4,135
1
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1
answer
158
views
Positive analytic semigroup
On $L^p(\mathbb{R}^+)$, we consider the following operator:
$$
Af:= f'',\qquad D(A):=\{u\in W^{2,p}(\mathbb{R}^+),u'(0)=0\}
$$
Now I want to know if this operator is a generator of a positive analytic ...
- 13
1
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1
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158
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Nonlinear elliptic problem involving the p-laplacian, Hölder inequality
I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan.
I have a problem understanding one step ...
- 73
1
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1
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672
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Fractional Sobolev spaces on the circle with a Littlewood-Paley characterisation
Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L_p(\mathbb R)$.
Here, $F$ denotes the ...
- 225
1
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1
answer
110
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Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$
Let $\Omega
\subset
\mathbb{R}^{N}$
be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is a Caratheodory function such that $g(x,t)=0$
for $t\leq0$
. Suppose that ...
- 135
1
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1
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432
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Does Trudinger inequality implies this critical Sobolev embedding?
Let $1<p<\infty$, $\mathrm{L}^p_s(\mathbb{R}^n)=J_s(\mathrm{L}^p(\mathbb{R}^n))$, where $J_s=(I-\Delta)^{-\frac{s}{2}}$, or $\mathscr{F}(J_sf)(\xi)=(1+|\xi|^2)^{-\frac{s}{2}}\hat{f}(\xi)$. And ...
- 1,616
1
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1
answer
264
views
Image of (right) inverse trace map $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$ dense in $H^1(\Omega)$?
Let $\gamma\colon H^1(\Omega) \to H^{\frac 12}(\partial\Omega)$ be the linear trace map which has a right continuous inverse $\xi\colon H^{\frac 12}(\partial\Omega) \to H^1(\Omega)$.
Is the image of $...
- 93
1
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1
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191
views
Getting an a priori bound on a nonlinear gradient term in PDE; how to adapt trick from $L^2$ case to $H^{-1}$ case?
I have the PDE
$$u_t(t) - \Delta f(u(t)) = 0$$
in $H^{-1}(\Omega)$ where $f$ is a nonlinear function.
Define $F(s) = \int_0^s f(s)$. Note that if $u_t(t) \in L^2(\Omega)$,
$$\frac{d}{dt}F(u(t)) = f(...
- 266
1
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1
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124
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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 \...
- 31
1
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1
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675
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First derivative of cut off function
I am working on proving the following: Let $\rho(x)= \frac{2}{2+x^2}$, $\theta >1$ (assumed integer here) and $B \subset H^1_{ul}$,(uniformly local Sobolev space), be any subset which is bounded in ...
- 159
1
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1
answer
1k
views
Does weak-* convergence in $W^{1,\infty}$ imply weak-* convergence in $L^\infty$?
Let $\Omega \subset \mathbb{R}^n$ be open and bounded.
What does weak-* convergence for a sequence of functions $\{f_k\}_{k \in \mathbb{N}}$ in $W^{1,\infty}(\Omega)$ mean? It seems to me that there ...
- 13
1
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1
answer
176
views
Classical fixed-point argument and invertible function
Let $n\in\mathbb{N}$ and $W^{1,\infty}(\mathbb{R}^n)=\lbrace f:\mathbb{R}^n\rightarrow \mathbb{R}^n : \text{ f is bounded and Lipschitz continuous } \rbrace$. Suppose $f\in W^{1,\infty}(\mathbb{R}^n)$ ...
- 237
1
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2
answers
226
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Logarithmic Sobolev embeddings
It is well-known that $H^{\frac d2}(\mathbb R^d)=W^{\frac d2, 2}(\mathbb R^d)$ is not included in $L^\infty(\mathbb R^d)$, but it seems that there are some logarithmic substitutes. Is it true for ...
- 16.2k
1
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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
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1
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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
1
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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
$$\begin{align*}
S_1 &= \left\{ \begin{...
- 119
1
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1
answer
170
views
A time dependent variational problem coming from a second order linear PDE
Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\...
- 597
1
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1
answer
374
views
Characterization of a subset of the Sobolev space $H^k(0,2\pi)$ in terms of Fourier series
Let $A:=\{ u \in H^k(0,2\pi): u^{(j)}(0)=u^{(j)}(2\pi) \mbox{ for } j=0,1,\ldots, k-1\}$, where $H^{k}(0,2\pi)\subseteq L^2(0, 2 \pi)$ is the Sobolev space of order $k$ on $(0, 2 \pi)$. Can we say ...
- 65
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1
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396
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Equivalence of Sobolev spaces for different metrics
Consider $M$ a manifold and $g_1, g_2$ two different Riemannian metrics. I want to know how the condition $|\nabla^{g_1,k}(g_1-g_2)|_{g_1}\leq C$ implies that the norms of $|\nabla^{g_1,i}u|_{T^{\...
- 156
1
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1
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130
views
Density on a specific functional space.
I have a question about density. It's probably trivial but I am just learning functional analysis so nothing is trivial to me. Here is my question. Let
$$
\mathcal{X}\colon=\mathcal{H}^1(0,1;\mathbb{...
- 11
1
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1
answer
216
views
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 ...
- 455
1
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1
answer
628
views
Continuity of a multiplication operator in fractional Sobolev space
Let $\Gamma$ be a regular boundary of a $C^{k,1}$ domain $\Omega$ and $H^s(\Gamma)$, $s\in(0,1)$, denote the fractional Sobolev space on $\Gamma$. Suppose I define a multiplication operator $M_\phi:H^...
- 363
1
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1
answer
165
views
Morrey condition (integral condition) and (local) Holder condition
Let $x \in \mathbb{R}^n$ and $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$)
$$\limsup_{r \to 0} r^{-\alpha \beta}\frac{...
- 839
1
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1
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143
views
Sobolev Embedding on Finite Tube
Let $B=[0,T]\times Y$, here $Y$ denotes a closed manifold. Suppose we use the product metric on this finite tube.
We can have the following inequality
$$\|s\mid_{t\times Y}\|_{L^2_1(Y)}\leq Const \|...
- 1,915
1
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1
answer
162
views
Regularity of integral kernel
Let $\Omega \subset \mathbb{R}^n$ be some open set.
If, for all $\psi \in L^2(\Omega)$ and some fixed integral kernel $k \in L^2(\Omega\times \Omega)$ and $\ell>0$, it is true that both
$\int_{\...
- 305
1
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1
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1k
views
Introductory text to Sobolev spaces and PDE's [closed]
I'm looking for a good introductory to Sobolev, preferably with an emphasis to their relationship to PDE's analysis.
I have only seen thus far Giovanni Leoni's "First Course in Sobolev Spaces" which ...
- 3,574
1
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1
answer
243
views
$L^\infty(0,T;X) \cap C([0,T];Y) \subset C([0,T];X)$ for $X \subset Y$ dense?
is the Inclusion stated in the title true? In my case the spaces (essentially) are $X = H^1(\Omega)$ and $Y = L^2(\Omega)$ for $\Omega \subset \mathbb{R}$ bounded. My first try was to show
$\lim_{t_1 ...
- 11
1
vote
1
answer
618
views
Condition to obtain a not compact embedding
I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ ...
- 277
1
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1
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514
views
$u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;H)$?
Let $V \subset H$ be a dense and compact embedding. Let $$\lVert u_n\rVert_{L^\infty(0,T;H)} + \lVert u_n \rVert_{L^2(0,T;V)} < C$$ where $C$ is independent of $n$. It follows that eg. $u_n \...
- 35
1
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1
answer
331
views
Weak convergence of a sequence
I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : $\int_{[...
- 185
1
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1
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353
views
A question about PDE argument involving monotone convergence theorem and Sobolev space
I'm reading this paper. In it there is the following argument (see page 240).
Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. $b(\...
- 266
1
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1
answer
367
views
Getting a comparison principle for parabolic equation when solution is not that smooth
Consider the solution $b(u) \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ with $u \in L^2(0,T;H^1)$ to
$$\frac{\partial}{\partial t}b(u) - \Delta u = f$$
where $b$ is continuous, increasing and locally ...
- 145
1
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1
answer
133
views
Nonintegrable inverse powers as distributions
I am working through Lieb/Loss's "Analysis", and have been stuck on one of the problems for a while;
Suppose we are on $\mathbb{R}^n$ and define $f(x) = |x|^{-n}$. This is not a locally integrable ...
- 13
1
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0
answers
58
views
duality of sobolev spaces. Representation of elements in the dual
I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
1
vote
0
answers
84
views
Does sets of positive capacity rule out constant functions?
Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by
\begin{align*}
\text{Cap}_{p}(K, U) :=
\inf \left\{
\int_U |\...
- 1,101
1
vote
0
answers
65
views
Fractional Sobolev embedding
Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these ...
1
vote
0
answers
98
views
Equivalence of Sobolev norms for smooth functions with compact support
Let $f\in C^\infty_c([0,1]^n)$, then we can extend it to a $1$-periodic smooth function $\tilde f$. We define the fourier transform (series) of $f$ ($\tilde f$):$$
\hat f(\xi):=\int e^{2\pi i x\cdot \...
- 435
1
vote
0
answers
174
views
Interpolation of Sobolev spaces with constraints
Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now,...
- 81
1
vote
0
answers
67
views
regularity convolution of a $L^2$ function with $W^{1,1}$ function [closed]
Let $u\in L^2(\mathbb R)$ and $w \in W^{1,1}(\mathbb R)$, we consider the convolution
$$u*v$$
Is it true that $w*u \in W^{1,2}(\mathbb R)$?
What regularity can we put on $w$ for this to be true?
- 73
1
vote
0
answers
51
views
Compact embeddings RKHSs into Sobolev Spaces
Let $\mathcal{H}$ be an RKHS over an open domain $\Omega \subseteq \mathbb{R}^d$. Are there conditions under which $\mathcal{H}$ can be compactly embedded in a Sobolev space $W^{s,p}(\Omega)$ for ...
1
vote
0
answers
93
views
Multivariate polynomial approximation
Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$.
Let $P_n$ is a space of polynomials with degree $n$. My question is what is the ...
- 61
1
vote
0
answers
78
views
Trace theorem for $L^2([0,1]; H^k(S^2))$
Consider a function $u$ in $L^2([0,1]; H^k(S^2))$ where $k$ is a positive integer.
Where would $u(0)$ live (or $u(r)$ for some fixed $r \in [0,1]$)? Is there a version of the trace theorem saying that ...
- 969
1
vote
0
answers
75
views
$T$ trace, then $Tg(u)=g(T(u))$ for all $u$ on $W^{1,p}$
The trace operator $T$ is defined for bounded domain $U$ with $C^1$ boundaries as the linear, continuous operator
$T: W^{1,p}(U) \rightarrow L^p(\partial U)$
such that
$$
Tu=u\;\text{ on }\partial U
$$...
- 11
1
vote
0
answers
210
views
Is this a well known space? Perhaps homogeneous Sobolev-like space?
The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...
- 197
1
vote
0
answers
123
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
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 11
1
vote
0
answers
109
views
$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$u' - \Delta u = 0 \quad\text{on $\Omega$}$$
$$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$
$$u(t=0)=u_0\quad\text{on $\Omega$}$$
where $\Omega$ is a bounded ...
- 93
1
vote
0
answers
145
views
Operator norm of linear functional $\varphi \mapsto \int_\Omega f\varphi$ with respect to different norms
Let $\Omega \subseteq \mathbb{R}^n$ be open. For some $f \in L^2(\Omega)$ consider the continuous linear functional $$T \colon C^\infty_c(\Omega) \to \mathbb{R}, \qquad T(\varphi) := \int_\Omega f \...
- 548