All Questions
18 questions
12
votes
0
answers
476
views
Are Sobolev trace spaces equal from both sides of the boundary?
Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
- 8,086
11
votes
3
answers
1k
views
Boundedness of the derivative of the trace of an H^1 function
As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter....
- 882
11
votes
1
answer
433
views
Best constant for a trace inequality
Having an open, simply connected set $\Omega \subset \Bbb{R}^N$ we may ask what is the best constant $C$ (if it exists) in the inequality
$$ \int_{\partial \Omega} u^2 \leq C\int_{\Omega} |\nabla u|^...
- 2,222
5
votes
1
answer
875
views
traces of sobolev spaces under additional assumptions
Let $p\in [1,\infty]$, $\Omega$ an open bounded domain with (smooth, if necessary) boundary $\partial \Omega$.
Is there a subspace $X\subset L^p(\Omega)$ - a simply describable space, ideally a ...
- 3,388
5
votes
1
answer
152
views
Trace spaces on convex polyhedra: compatibility conditions at edges
Let $\Omega$ be a convex polyhedron in $\mathbf{R}^3$ with boundary $\partial\Omega$ consisting of $N$ polygons $\{\Gamma_j\}_{j=1}^N$. It's well known that in general
$$ H^s(\partial \Omega) \neq \...
5
votes
0
answers
101
views
A bounded extension operator
Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
- 81
5
votes
0
answers
927
views
Trace Theorem for $p=\infty$
I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one (...
- 59
3
votes
1
answer
902
views
Fractional Sobolev spaces and extension by zero
The all-wise Wikipedia suggest (https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero) that for $s > 1/2$ any function from $H^s_0(\Omega)$ can be extended by zero
to whole $\mathbb{R}^n$ (...
- 51
3
votes
1
answer
752
views
A distributional normal derivative for functions in $H^1(\Omega)$
Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...
- 266
3
votes
0
answers
82
views
Compatibility between the source and the boundary condition for an Helmholtz-type equation
Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
- 943
3
votes
0
answers
304
views
Traces of fractional Sobolev spaces $W^{s,p}$ with $0<s<1/p$
I've stumbled upon a problem involving the trace of a function in a fractional Sobolev space of the form $W^{s,2}(H)$, where $H$ is a half-plane in $\mathbb{R}^2$. Would it be possible to define a ...
- 31
3
votes
0
answers
74
views
Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$
Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
- 251
2
votes
1
answer
766
views
reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
- 5,380
2
votes
2
answers
241
views
A Characterization of the traces of functions in $W^{1,2}$
I have a question about the traces of functions in $W^{1,2}$.
Let $D$ be a connected open subset of $\mathbb{R}^d$.We denote $W^{1,2}(D)$ by
\begin{align*}
W^{1,2}(D)=\{f \in L^{2}(D,dx) \mid \...
- 721
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
1
vote
1
answer
162
views
Why is the relative trace of Sobolev norms finite?
I am reading the 2009 Paper on Effective Equidistribution by Einsiedler, Margulis and Venkatesh (EMV). I do not understand Section 5.3 on the proof of (3.10). They want to prove that the relative ...
- 429
1
vote
1
answer
144
views
Does this time-dependent trace space have a name?
This question is a follow up to this question.
Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u_d:\partial\Omega \to \mathbb{R}$ be in $H^{1/2}(\...
- 142
1
vote
0
answers
72
views
Trace of $u$ on bottom edge of a square if $u_x=0$ inside the square
I want to show that:
Let $\Omega =(0,1)\times (0,1)$. For $u \in H^1(\Omega)$, if $u_x=0$ a.e. in $\Omega$, then the trace of $u$ on bottom edge $y=0$, i.e., $u\left|_{y=0}\right.$, is a constant.
...
- 21