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Vector field connecting two points

I'm now working on somehow an inverse problem of an ODE: Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t). Now there is a ...
Sqr's user avatar
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80 views

Relationship between two minimization problems

Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
Cauchy's Sequence's user avatar
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143 views

A Poincaré inequality holds for $p>2$ but fails for $p\leqslant 2$

I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022). Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset ...
Javier's user avatar
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55 views

Time regularity vs space regularity for parabolic PDE

Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...
Azam's user avatar
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109 views

A Lipschitz function induced by the infimum of the length of curves

Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of ...
Javier's user avatar
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211 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}...
Muniain's user avatar
  • 101
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37 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 ...
Isaac's user avatar
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67 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 ...
NullOfMatrix's user avatar
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119 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 ...
Elio Li's user avatar
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149 views

Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$

I am looking at Corollary 1. in p.244-245 of the book "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations" (1996) by Thomas Runst Winfried ...
Isaac's user avatar
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78 views

Sobolev trace inequality with exterior domains

Let $x_1\in \mathbb{R}^n$, $n\geq 3$, $\Omega=\mathbb{R}^n\backslash B_1(x_1)$, define $D_{\Omega}$ by taking the closure of $C_{c}^{\infty}(\overline{\Omega})$ under the norm \begin{align*} \|u\|_{D_{...
Davidi Cone's user avatar
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142 views

Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?

Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
Dokem's user avatar
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103 views

Extension for fractional Sobolev spaces, s>0

In their paper, Fractional Sobolev extension and imbedding, the author describes all extension domains for $s \in (0,1)$ -- meaning spaces functions in which are not required to have weak derivatives. ...
Athere's user avatar
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152 views

Concentration compactness lemma and the best Sobolev constant

It is well known that the best Sobolev constant can be achieved on $\mathbf{R}^n$. More precisely, we have the following theorem (A): Let $\frac{1}{q}=\frac{1}{2}-\frac{1}{n}$, $$S=\inf\limits_{{u\in ...
sorrymaker's user avatar
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21 views

Estimatives for elliptic systems involving the laplacian

Considering the problem \begin{equation} \left\{ \begin{array}[c]{11} \Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\ \Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\ \end{...
Bruno Mascaro's user avatar
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165 views

Compact embedding of Lipschitz continuous functions

Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions ...
Delio Mugnolo's user avatar
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129 views

Bounding trace operator from below

In a paper, I've read the following thing. Here $\Omega$ is a smooth domain From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
bobinthebox's user avatar
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1 answer
213 views

'Diamagnetic' inequality for negative Sobolev spaces

Let us look at the subspace of smooth complex functions of $L^2(\mathbb{R}^n,\mathbb{C})$, call $H^s$ the Sobolev spaces. By the diamagnetic inequality $\lvert \nabla \lvert\psi\rvert\rvert (x) \le \...
user avatar
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106 views

Extension of a Hilbert basis

The picture below is taken from this paper: http://real.mtak.hu/22877/. The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
Gustave's user avatar
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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 ...
Athere's user avatar
  • 93
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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 ...
Gustave's user avatar
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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{...
Guilherme's user avatar
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97 views

Uniform bound on $\lVert \chi_{\{u_n=0\}}\rVert_{W^{s,p}(\Omega)}$ for a bounded sequence $u_n$ in $H^1_0(\Omega)$?

Suppose I have a sequence $u_n \to u$ in $H^1_0(\Omega)$ on a smooth and bounded domain. For some $p>1$ and $s \in (0,\frac 12)$, is it possible to estimate the norm of the characteristic function ...
StopUsingFacebook's user avatar
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1 answer
545 views

Orlicz–Sobolev spaces

Let $A$ be an N-function and suppose that $$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty. $$ We denote by $\widehat{A}$ an N-function equal to $A$ near infinity and $\widehat{...
deval si-mohamed's user avatar
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227 views

Negative Sobolev norm of non-zero mean non-periodic function on bounded space

The usual formulation of $H^{-1}$ norm for a zero-mean periodic function on some domain $\Omega\in\mathbb{R}$ is as follows: $\|f\|^2_{H^{-1}}=\sum\limits_{k\in Z, k\neq 0}\dfrac{\hat{f}^2_k}{k^2}$, ...
mystupid_acct's user avatar
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0 answers
72 views

Sobolev embedding for a specific family of weighted Sobolev spaces

Consider the weighted Sobolev-type space $$ W_\alpha:=\{f\in L^2(0,\infty):\hbox{id}^\alpha\cdot f'\in L^2(0,\infty)\}. $$ Are there any known embeddings? Ideally, I am looking for an embedding of the ...
Delio Mugnolo's user avatar
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96 views

Does adding a compact operator change the symbol of a pseudodifferential operator?

Suppose $X$ is a non-compact manifold. Let $P$ be an order-$0$ pseudodifferential operator on $X$ and $f:L^2(X)\rightarrow L^2(X)$ a compact operator. I'm wondering: 1) Is $P + f$ always a ...
geometricK's user avatar
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0 answers
81 views

Differential operator and equivalence

Here is the problem: I have a certain PDE and there is the nonlinear terme $h$, I have as data: $f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$ Now on consider the fnction $$h(...
Gustave's user avatar
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0 answers
308 views

Invertible operator

We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$ We hope to prove that $T$ is invertible if and only if $L = n\pi $. and for this ...
Gustave's user avatar
  • 617
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0 answers
59 views

Differential operator

One define the operator $T$ as :$$T: = (I - {{{\partial ^2}} \over {\partial {x^2}}}):H_0^1(0,L) \cap {H^2}(0,L) \to {L^2}(0,L) $$ let $f \in H_0^2(0,L) \cap {H^4}(0,L)$. What can we say about ${T^{ - ...
Gustave's user avatar
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0 answers
89 views

If $H$ is the closure of the set of solenoidal smooth vecor fields in $L^2$ and $P_H$ denote the orthogonal projection onto $H$, then $P_HH_0^1⊆H_0^1$

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$ and $$H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\...
0xbadf00d's user avatar
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0 answers
107 views

Is $(u\cdot\nabla)v\in H^1$, if $u,v\in H^2$?

Let $d\in\left\{2,3\right\}$ with $\Lambda\subseteq\mathbb R^d$ be bounded and open with $\partial\Lambda\in C^1$ In Lemma 6.1 of Navier-Stokes Equations and Nonlinear Functional Analysis by Roger ...
0xbadf00d's user avatar
  • 167
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0 answers
215 views

Intersection of weighted Sobolev spaces

Consider the Sobolev spaces with $p=2$, defined for $s \in \mathbb{R}$ as \begin{equation} W^{s} = \left\{ u \in \mathcal{S}', \ (1 + \lvert \cdot \rvert^2)^{{s}/{2}} \widehat{u} \in L_2 \right\}. \...
Goulifet's user avatar
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0 answers
378 views

compact injection

Put: $D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$ Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...
Fadil Kikawi's user avatar
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0 answers
64 views

Approx the jump point of a $BV$ function from both hand side

Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as $$ u(x)= \begin{cases} 0,&\text{ if }x\in(-1,0)\\ 1,&\text{ if }x\in(0,1) \end{cases} $$ Clearly, we have $u\in ...
JumpJump's user avatar
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0 answers
470 views

Derivatives of Mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following: Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
ghjdnkmttrasda's user avatar
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0 answers
54 views

Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$

Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin. Is it true that $$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...
JamesCa's user avatar
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0 answers
187 views

About equivalence of two fractional Sobolev/Hilbert spaces

Let $(\varphi_k, \lambda_k)$ be the eigenelements of the Neumann Laplacian. It's possible to define a space $$H(\Omega) = \{ u \in L^2(\Omega) \mid \sum_{k \geq 1}\lambda_k^{\frac 12}|(u,\varphi_k)_{L^...
C_Al's user avatar
  • 251
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0 answers
200 views

Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$

We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$ I want to show for $r \geq 2$ that $$\...
delimit's user avatar
0 votes
1 answer
217 views

Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary [closed]

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form $$|u(t)|_{L^p} \leq ...
TomJoseph's user avatar
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0 answers
606 views

$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$

This question stems from the proof of Theorem A.1 on page 425 of this paper. Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in $C^0(...
riem's user avatar
  • 266
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0 answers
160 views

Is this function in the weighted Sobolev space $H^{2,-s}$?

I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order $...
Sue's user avatar
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0 answers
214 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
Miguel's user avatar
  • 101
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0 answers
362 views

Gradient of the energy functional in $H^{1,2}$-norm

I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy ...
Orbicular's user avatar
  • 2,935
-1 votes
1 answer
187 views

Limit of a function in a weighted Sobolev space

I have a function $f(x)$ in the space $H^{2,-s}(\mathbb{R}^3)$; have this limit sense $$\lim_{|x-y|\to 0} f(x)$$ ? ($y$ is a fixed point) If i have $f$ in $H^2$ I can say that $$\lim_{|x-y|\to 0} f(x)=...
Sue's user avatar
  • 25
-1 votes
1 answer
153 views

Sobolev estimates $\|\nabla\phi\|_{\infty}\leq C\|\phi\|_{H^2}$

This is a cross post in continuation to this question on Mathematics Stack Exchange. I wanted to know if this inequality holds true in two or three dimensions, $\|\nabla\phi\|_{L^{\infty}(\Omega)}\leq ...
Mainak's user avatar
  • 101
-1 votes
0 answers
53 views

convergence of convolution in Bochner space

I want to prove a well-known fact in $L^p(R^n)$ namely that, the convolution of an element in $L^p$ with an element of $L^1$ is in $L^p$ let: if $u∈L^p (R;X) , f∈L^1 (R)$ and $X$ is Separable and ...
Alucard-o Ming's user avatar
-1 votes
1 answer
86 views

how take weak derivative of norms in hilbert spaces?

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); ...
Alucard-o Ming's user avatar
-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
user895874's user avatar
-2 votes
1 answer
216 views

Inverse of Sobolev interpolation inequality : $\lVert u \rVert_2 \lVert \Delta u \rVert_2 \leq C\lVert \nabla u \rVert_2^2$?

If $u : \mathbb{T}^3 \to \mathbb{R}$ is a smooth function on the $3$-dimensional torus $\mathbb{T}^3$, I wonder it is possible to reverse the Sobolev interpolation inequality in the sense that \begin{...
Isaac's user avatar
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