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Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains

On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$. I'm interested ...
user2103480's user avatar
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 |\...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
83 views

3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$

Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$. Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
Himanshu Garg's user avatar
2 votes
1 answer
75 views

How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$

Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$. Assume ...
Arghya kundu's user avatar
0 votes
0 answers
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
3 votes
1 answer
187 views

Is this property preserved under weak$^*$ convergence?

Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
Cauchy's Sequence's user avatar
3 votes
0 answers
90 views

Sobolev embedding on a compact manifold without boundary

I am reading M. E. Taylor, "Partial Differential Equations III", Second Edition, Springer-Verlag, New York, (1996). In chapter 13, section 2, in Prop. 2.3 and Prop. 2.4, one finds the ...
Azam's user avatar
  • 311
2 votes
0 answers
103 views

A question from a proof of an inequality in Sobolev space $W^{1,1}$

I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot: Here is what I did: $$-u(x)=u(y)-u(x)=\...
Bogdan's user avatar
  • 1,759
4 votes
3 answers
308 views

Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$

Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
Bogdan's user avatar
  • 1,759
3 votes
1 answer
376 views

A more general product rule for weak derivatives?

Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
Bogdan's user avatar
  • 1,759
2 votes
0 answers
92 views

Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries

For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$ \begin{...
Arghya kundu's user avatar
2 votes
1 answer
117 views

Special density on $L^2$

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
Bogdan's user avatar
  • 1,759
6 votes
1 answer
246 views

The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by $$Lu = \partial_i(a^{ij}...
MathsGoose's user avatar
3 votes
0 answers
167 views

Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate

If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
Cauchy's Sequence's user avatar
5 votes
2 answers
364 views

Euler-Lagrange equations for minimizer of energy with indicator function

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
BBB's user avatar
  • 93
8 votes
0 answers
177 views

Understanding spaces of negative regularity

I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here. Let $C^k(\mathbb{R}^n$) be the ...
CBBAM's user avatar
  • 721
3 votes
0 answers
147 views

Embeddings of Bochner-Sobolev spaces with second time derivative

NOTE: I also asked this question here in MSE. In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
MathsGoose's user avatar
0 votes
0 answers
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
  • 311
3 votes
0 answers
128 views

Image of trace operator on $W^{2,1}(\mathbb{R}^2)$

It is known that for a domain $\Omega\subset \mathbb{R}^2$ with $C^1$ boundary $\partial\Omega$, that the trace operator is bounded and surjective from $T:W^{1,1}(\Omega)\to L^1(\partial\Omega)$. For ...
vmist's user avatar
  • 989
2 votes
0 answers
56 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 ...
BBB's user avatar
  • 93
9 votes
1 answer
639 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 ...
Zhang Yuhan's user avatar
1 vote
1 answer
137 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 ...
Bogdan's user avatar
  • 1,759
4 votes
1 answer
147 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(\...
MeS's user avatar
  • 41
3 votes
1 answer
296 views

Weighted Lebesgue space with exponential weights: smoothing effect and properties

I am researching whether there are weighted Lebesgue spaces of the type $$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$ ...
Ilovemath's user avatar
  • 677
2 votes
1 answer
102 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-...
alia's user avatar
  • 23
2 votes
0 answers
164 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 ...
Isaac's user avatar
  • 3,477
6 votes
1 answer
331 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) ...
Isaac's user avatar
  • 3,477
8 votes
1 answer
496 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 $u\in C^{\infty}_0((-1,1))$?
Ali's user avatar
  • 4,135
2 votes
1 answer
620 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 ...
Arghya kundu's user avatar
3 votes
1 answer
338 views

On a Poincaré inequality with weight

Let $\Omega$ be a bounded convex (non-empty) open subset of $\mathbb{R}^n$ ($\Omega$ can be as smooth as you like). Let also $p, q > 1$ be conjugate exponents. Is it true that there exists a ...
Romain Gicquaud's user avatar
4 votes
2 answers
781 views

Is there any bilinear Poincaré/Sobolev inequality?

Is the following, I call it bilinear Poincaré inequality, true? Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
Hao Yu's user avatar
  • 185
2 votes
0 answers
160 views

Understanding the Bochner space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ in terms of the Fréchet derivative

In the context of linear parabolic equations, the Sobolev space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ appears all the time. Here, $U$ is some bounded region of $\mathbb{R}^n$ and $1<p,q<\infty$. ...
Isaac's user avatar
  • 3,477
2 votes
0 answers
153 views

Riesz’s representation theorem in a weak form

Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$ \begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
Davidi Cone's user avatar
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 ...
BBB's user avatar
  • 93
0 votes
1 answer
102 views

Limit of minimizers of a class of functionals

Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional $$ \mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx $$ where $ h>0 $ is a parameter and $ ...
Luis Yanka Annalisc's user avatar
0 votes
1 answer
154 views

Finite dimensionality of a subspace

Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be subspace of $Y$ such that for any $\phi \in X$ there holds: $$ \...
Ali's user avatar
  • 4,135
3 votes
0 answers
102 views

Can Sobolev space be characterized by spectral decomposition?

Consider a homogeneous Carnot group $\mathbb{G}$ with step $r$. Let $X_1,\cdots,X_m$ be the first layer of its Lie algebra. Denote by $\mathcal{L}=-\sum_{i=1}^m X_i^2$ the sub-Laplacian on $\mathbb{G}$...
Houa's user avatar
  • 561
5 votes
1 answer
216 views

Bounds on dimension of a subspace

Let $I=(0,1)$ and let $C>1$ be a constant. Let $L^2(I)$ and $H^1(I)$ be the standard Sobolev spaces on $I$. Suppose that $U$ is a subspace of $H^1(I)$ with the additional property that: $$ \| u\|_{...
Ali's user avatar
  • 4,135
0 votes
0 answers
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
  • 1
3 votes
0 answers
88 views

Using a maximum principle to deduce regularity

Suppose $\Omega \subset \mathbb{R}$ is an bounded domain and that $u \in C(0,T; H^{2}) \cap L^{2}(0,T; H^{3})$ where $T >0$. Consider the PDE on $\Omega \times [0,T]$ $$ \partial_{t}u = a_{1}(x,t) \...
duelspace's user avatar
  • 131
3 votes
0 answers
209 views

Interpolation between Sobolev spaces

In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by $$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$ where $D^sf$ is defined by the Fourier transform $$(D^...
kuuga's user avatar
  • 71
1 vote
0 answers
52 views

A question of interpolation space on homogeneous Carnot group

Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows: A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
pxchg1200's user avatar
  • 287
2 votes
0 answers
259 views

Research in analysis of PDEs

I am currently figuring out what topic to work on for my undergraduate thesis and was able to narrow it down to mathematical analysis. As of now, I have two main options: a thesis working on Sobolev ...
Gab Romero's user avatar
2 votes
1 answer
154 views

Function monotony between [0,T] and $L^2$

Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...
Bogdan's user avatar
  • 1,759
3 votes
1 answer
466 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
Zac's user avatar
  • 161
6 votes
1 answer
173 views

Sobolev space is spanned by distributions supported on half-lines?

I asked this question on Mathematics Stack Exchange previously. This seems to be a very basic property of Sobolev spaces, but I wasn't able to find a proof for it. For any $s \leq 1/2$, $$H^s(\mathbb{...
Tian Xia's user avatar
2 votes
0 answers
150 views

Reference for weighted Sobolev spaces

I'm looking for a comprehensive reference illustrating, from the ground up, the basics of weighted Sobolev Spaces on Lipschitz domains (this case should be included, but I don't need less than it). ...
Lilla's user avatar
  • 235
3 votes
1 answer
741 views

About radial Sobolev inequality (Strauss Lemma)

As shown in Strauss: Existence of solitary waves in higher dimensions, Strauss introduces the Stauss lemma. Precisely speaking, we have the following theorem: Theorem Let $N \ge 2$, every radial ...
Tao's user avatar
  • 429
3 votes
0 answers
324 views

Would you help me to find this expression?

I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate $$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$ Where $\Omega$ is an ...
Silvinha's user avatar
1 vote
0 answers
93 views

$H^s$ norm of dispersive semigroup

The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm $$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle} u(t,x)...
Mr. Proof's user avatar
  • 159

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