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Questions tagged [trace-map]

Questions about the trace map on Sobolev spaces, which maps a function on a domain to its boundary values, and generalizations or related concepts. Consider using [sobolev-spaces] as well. For questions about the trace of a matrix or other meanings of trace, please use [trace] instead.

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On the injectivity of the trace operator

Let $\Omega \subset \mathbb{R}^N$ be an open bounded domain with $\partial \Omega$ be class $C^1$. I have to show that if $f \in H^1(\Omega)$ and $g \in W^{1,\infty}(\Omega)$ then the product $fg$ is ...
Contrad's user avatar
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Trace theorem in $H^2(\Omega)$ and the affective restriction on the boundary

In Theorem 2.7.4 from Kesavan Topic in Functional Analysis 2003 says that there exists a map $\gamma = (\gamma_0, \gamma_1)$ from $H^2(\Omega)$ to $(L^2(\Omega))^2$ such that If $v \in C^\infty(\...
Lucas Linhares's user avatar
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Restriction of a compactly supported function on a bounded domain in a surface

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and let $P_k$ a plane of dimension $1\leq k<N$ in $\mathbb{R}^N$. Denote by $\sigma_k$ the surface measure in the surface $\Omega_k = \Omega\...
Lucas Linhares's user avatar
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How is the trace theorem used in practice?

Here is what I know. Let $\Omega \subset \mathbb{R}^n$ be bounded and $\partial \Omega$ be $C^1$. If $f \in C(\Omega)$ then there, if $T$ is the classical trace operator, $Tf = f|_{\partial \Omega}$ ...
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Is there a way to prove the theorem of Trace as in Evans, in the picture below, using only weak derivatives?

Is there a way to prove the theorem of Trace as in Evans, in the picture below, using only weak derivatives? I mean, without using the divergence theorem?
Silvinha's user avatar
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There is no continuous functional $T:L^p(\Omega)\rightarrow L^p(\partial \Omega)$, with $\Omega\subset\mathbb{R}^n$? (As in trace theorem)

Why it is not possible to define a continuous functional such that \begin{equation} \begin{split} T\colon L^p(\Omega)& \rightarrow L^p(\partial\Omega)\\ u&\mapsto Tu=u_{|_{\partial \Omega}}...
Silvinha's user avatar
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Trace spaces for Sobolev functions

I have a question regarding the definition of general trace spaces (e.g. from the book "Elliptic problems in domains with piecewise smooth boundaries" by Nazarov and Plamenevsky (Chapter 2.2....
snape1234's user avatar
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How to associate trace on finite fields with linear algebraic trace

I am familiar with the partial trace operator that one deals with in Quantum computing on density matrices. That would be sum of diagonal elements. I was reading a section on finite fields and ...
Zee's user avatar
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Positive part of trace of a Sobolev function

Suppose that $\Omega$ has a smooth boundary, $u\in H^1(\Omega)$. I want to know whether the following holds: $$ tr(u^+)=tr(u)^+ $$ where $tr$ denotes the trace of a Soboelv function. In other words, ...
zik2019's user avatar
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Divergence theorem in Sobolev spaces

Let $\Omega\subseteq \mathbb{R}^N$ be a regular domain (a bounded open set with $C^1$ boundary) and let $X$ be a $W^{1,1}(\Omega,\mathbb{R}^N)$ be a Sobolev vector field. I want to prove the following ...
Kandinskij's user avatar
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Continuous functions in Sobolev spaces

Let $W^{k,p} (\Omega)$ be a Sobolev space, $\Omega \subset \mathbb{R}^N$. Formally, $W^{k,p}(\Omega)$ consists of equivalence classes of functions with finite Sobolev norm. Two functions, $f$ and $g$, ...
mathslover's user avatar
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Reference for extension operators on Sobolev spaces

Let $\Omega \subset \mathbb{R}^d$ be a domain with Lipschitz boundary and $W_p^k(\Omega)$ be the Sobolev space of functions over $\Omega$ for which weak derivatives up to order $k$ exist and have ...
sudeep5221's user avatar
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How are higher order boundary conditions handled in a weak formulation?

I am reading Gazzola et al's book and there for a space $H^{m}(\Omega)$ they seem to allow boundary conditions that are of degree as high as $2m-1$. How is this reconciled with the fact that ...
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Exercise 3, Section 3.7 of Hoffman’s Linear Algebra

Let $V$ be the space of all $n \times n$ matrices over a field $F$ and let $B$ be a fixed $n \times n$ matrix. If $T$ is the linear operator on $V$ defined by $T(A) = AB - BA$, and if $f$ is the trace ...
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Trace of a Sobolev-Function near the boundary when shrinking tubes

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, $\delta>0$ and $x_0\in\partial\Omega$ fixed. Moreover let $Q_\delta(x_0)$ be a cube with sidelength $\delta$ around $x_0$ and define ...
Ferdi96's user avatar
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The trace theorem for functions in $H^{1/2}(\Omega)$

In my textbook, it said that we have the trace operator on Sobolev space like this: (Suppose $\Omega$ is a nice domain in $R^d$) \begin{equation*} H^{s}(\Omega) \hookrightarrow H^{s-\frac{1}{2}}(\...
Openminded's user avatar
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Founding bounds on a certain expression

Suppose we have a bounded domain $\Omega$ with a boundary $\Gamma$. The space $L^2(\Omega)$ is equipped with the usual norm and inner product $|| \cdot||$ and $(\cdot , \cdot)$. I was working on a ...
Tarek Acila's user avatar
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1 answer
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Trace of continuous function is restriction on boundary.

I was reading a theorem about the trace of a function (taken from Evans' book on PDEs). In the proof, I don't understand the last paragraph. Why the uniform convergence of $u_m$ to $u$ implies $Tu=u|_{...
edamondo's user avatar
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Reference request: Lebesgue/Sobolev spaces on the boundary

I am interested in the Boundary Lebesgue/Sobolev/Besov Spaces $L^p(\partial\Omega;\mathcal{H}^{N-1}), \ W^{k,p}(\partial\Omega),\ B^{s,p}(\partial\Omega)$ where $\Omega\subseteq\mathbb{R}^N$ is a ...
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Inverse of trace operator. (Extending a function on $\partial \Omega$ to $\Omega$)

Let us consider a smooth function $u$ defined on some open set containing $\bar{\Omega}$ which satisfies $u=g$ on $\partial \Omega$ when $\Omega$ is Lipschitz (You may assume that $\Omega$ is an ...
CHO's user avatar
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2 votes
1 answer
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Right inverse of trace operator

I'm studying the trace space on a smooth boundary $\Gamma$, namely $H^{\frac{1}{2}}(\Gamma)$. Let $\gamma$ be the classical trace operator $\gamma: H^1(\Omega) \rightarrow H^{\frac{1}{2}}(\Gamma)$. It ...
bob_bill's user avatar
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1 answer
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Seeking references regarding Zero Extension of Sobolev ($W_0^{1,2}$) functions on a compact domain on $C^\infty$ Riemannian manifold

I am looking for a reference of the well-known fact due to P. Jones and A. P. Calderon that a Sobolev function $f \in W^{1,2}_0(\Omega),\ \Omega \subset \mathbb{R}^n$, with the Dirichlet boundary ...
Igor O's user avatar
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2 answers
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Is there trace operator for periodic functions?

I know that for a smooth domain $\Omega$ we can build a trace operator $\gamma : H^s(\Omega) \to \prod_{0\leq j \leq s}H^{s-j-\frac{1}{2}}(\partial \Omega)$. In particular it has a right inverse which ...
cadiot matthieu's user avatar
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108 views

Trace property of $H_0^1(\Omega)$

Do we have: $$ H_0^1(\Omega)=\{f\in H^1(\Omega): \gamma f=0\}, $$ while $\gamma:H^1(\Omega)\rightarrow H^{1/2}(\partial \Omega)$ is the trace map? The teacher says that it's true in the course, ...
DreamAR's user avatar
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Bounding from below the trace operator

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
3 votes
2 answers
189 views

Assumption on bounded open in trace theorem- Sobolev Space

I have a problem in the proof of the theorem $$T(u)=0\implies u\in W_0^{1,p}(U).$$ Assume $U\subset \Bbb R^n$ bounded open and $\partial U$ is $C^1$. Suppose $u\in W^{1,p}(U)$. My question: Why can I ...
Domenico Vuono's user avatar
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1 answer
225 views

Optimization of trace of a matrix

I want to solve the following optimization problem , if R is a positive semi-definite matrix and D is diagonal matrix , both of size T x T then we should solve ${\underset{D}{min}}$ trace[$(R + D)^{\...
PHANI RAJ's user avatar
2 votes
0 answers
68 views

Surjectivity of trace operator for intersection of two (fractional) Sobolev spaces

Let $W^{k,p}$, for nonnegative integer $k$ and real number $1\leq p\leq\infty$ denote the usual Sobolev space. If we replace $k$ by a noninteger $s\geq 0$, then $W^{s,p}$ denotes the Bessel potential ...
Nik Quine's user avatar
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6 votes
1 answer
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Characterization of functions in the Sobolev space $H_0^2(U)$ as zero trace functions in $H^2(U)$

Firstly, I am wondering if there exists a trace operator $$T:H^2(U)\rightarrow L^2(\partial U)$$ such that it satisfies analogous properties to that of the usual trace operator for functions in $W^{1,...
Akerbeltz's user avatar
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2 votes
0 answers
241 views

A generalization of Green-Gauss divergence theorem to Sobolev functions on sets of finite perimeter

Let $n\ge 2$. Let $u \in W^{1,1}(\mathbb{R}^n)$, i.e. $u \in L^1(\mathbb{R}^n)$ and its distributional gradient is represented by an element of $L^1(\mathbb{R}^n;\mathbb{R}^n)$. By the Sobolev ...
Bob's user avatar
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-1 votes
1 answer
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About the trace inequality in Sobolev space [closed]

Given $\Omega$ is a region with Lipschitz domain and $1\leq p\leq +\infty$. Prove that there exists a constant $C>0$ such that $$ \Vert u\Vert_{L^p(\partial\Omega)} \leq C \Vert u\Vert_{L^p(\Omega)...
Bakkune's user avatar
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1 answer
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Regarding Trace of a function in $W^{1,p}$

By Trace theorem it implies that if $u\in W^{1,p}(\Omega)$ then $\text{Tr}(u)\in L^p(\partial \Omega)$. Now, if $\text{Tr}(u)\in L^p(\partial \Omega)$ can it be said that $u\in W^{1,p}(\Omega)$. Are ...
Arun's user avatar
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2 votes
1 answer
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Bounded Trace Sobolev Embedding

Just a sanity check here: all the references I've seen concerning Sobolev spaces discuss Sobolev trace embeddings of the form $$\|u\|_{L^q(\partial\Omega)}\le C\|u\|_{W^{1,p}(\Omega)}$$ where $\Omega$ ...
Fozz's user avatar
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2 votes
1 answer
515 views

Sobolev spaces with respect to divergence and their properties

Let $n \in \mathbb{N}$, $\Omega$ a non-empty bounded open set of $\mathbb{R}^n$ with Lipschitz boundary and $p \in [1,\infty]$. Define $$V_p:=\bigg\{\overrightarrow{q}\in L^p(\Omega;\mathbb{R}^n) \mid ...
Bob's user avatar
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1 vote
1 answer
206 views

Question about the trace map

Let k be a field and $f:M_n(k)\to k$ be linear transformation such that $f(AB)=f(BA)$ for all $A,B \in M_n(k)$. Show that f is a scalar multiple of trace map. I thought that it might be the ...
sayan Ghosh's user avatar
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0 answers
215 views

Equivalent Definitions of Field Trace of Galois Extension

When studying trace and norm of finite Galois extensions, I stumbled upon two (for Galois extensions) equivalent definitions. $Tr_{L|K}(a) = tr(\phi_a)$ with $\phi_a: L \to L, x \mapsto ax$ $Tr_{L|K}(...
Polly Nomial's user avatar
3 votes
2 answers
468 views

Higher boundary regularity theorem in Evans for elliptic equation

I have difficulty with the proof of this theorem: Let $m$ be a nonnegative integer, and assume $a^{ij},b^i,c\in C^{m+1}(\overline{U})$, for $(i,j=1,...,n)$ and $f\in H^m(U)$. Suppose that $u\in H^1_0(...
Domenico Vuono's user avatar
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0 answers
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How to prove that this cartesian product of subspaces is closed.

Let $l_0<l_1<l_2<l_3\in \mathbb{R}$, $I_1=(l_0,l_1),\, I_2=(l_1,l_2)$ and $I_3=(l_2,l_3)$. Define the spaces \begin{equation*} H^2_{l_0}=\{u\in H^2(I_1):u(l_0)=u'(l_0)=0\}\\ H^2_{l_3}=\{w\in ...
CharlesXavier's user avatar
2 votes
1 answer
481 views

Right-inverse of trace extension operator

Let $\Omega\subset\mathbb{R}^n$ be bounded with Lipschitz boundary. Then we know that there exists a bounded linear trace operator $T:W^{1,1}(\Omega)\to L^1(\partial \Omega)$. Howerever I want to know ...
L.P.'s user avatar
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0 answers
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Equality of traces of two matrices

I was going through Boyd's EE363 HW 4 . In page 11,I found this I dont understand why the 2 expressions involving trace (in the MSE equation ) are equal to each other . $\Sigma_{x}$ is postive ...
phani raj's user avatar
4 votes
1 answer
277 views

A doubt on the Sobolev space $W_0^{1,p}(\Omega)$

Let $\Omega \subset \Bbb R^d$ be open and $1\leq p<\infty$. Recall that $W_0^{1,p}(\Omega)$ is the closure of $C_c^\infty(\Omega)$(smooth function with compact support in $\Omega$) in $W^{1,p}(\...
Guy Fsone's user avatar
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4 votes
1 answer
213 views

$W_0^{1,p} \cap W^{1,q} = W_0^{1,q}$?

Let $\Omega \subset \mathbb R^n$ be an open set. We denote by $W^{1,p}(\Omega)$ the usual Sobolev spaces and $W_0^{1,p}(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega)$. Let $1 \le ...
gerw's user avatar
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1 vote
0 answers
184 views

Convergence of normal derivative in Sobolev space $H^1$

Let $\Omega$ be a bounded domain, $u_k\in H^1(\Omega)$, $k=1,2,..$ and the normal derivative $\frac{\partial u_k}{\partial \mathbf{n}}\in L^2(\partial\Omega)$. As $k$ goes to infinity, $u_k$ converges ...
Hui Zhang's user avatar
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1 vote
1 answer
272 views

Number of solutions of system of linear equations over finite field

Suppose $F=GF(2^8)$. Let $u_1,u_2\in K=GF(2^4)$ be linearly independent elements. The functions $x\mapsto tr_n (u_ix)$ are linear functions from $F\to GF(2)$, $i=1,2$. Here $tr_n$ denotes the absolute ...
zermelovac's user avatar
1 vote
0 answers
74 views

On the relation between $H^{1/2}(\partial Ω)$ and $\left(\ker(\text{tr})\right)^\bot$

This is a follow up to my question The Sobolev Space $H^{1/2}(∂Ω)$ as the Quotient Space $H_1(Ω)/\ker(\text{tr})$. Whilst this is another question related to the same space, I believe it focuses on ...
Jeremy Jeffrey James's user avatar
1 vote
1 answer
862 views

The Sobolev Space $H^{1/2}(\partial \Omega)$ as the Quotient Space $H^1(\Omega)/\ker(\text{tr})$

In both questions Reference request: norm of the image of a bounded linear operator and The Sobolev Space $H^{1/2}$, the Sobolev space $$H^{1/2}(\partial Ω) = \{ u ∈ L^2(\partial Ω) \;|\; ∃ \tilde u ∈...
Jeremy Jeffrey James's user avatar
1 vote
0 answers
45 views

Which space is $L^1(\partial\Omega)$?

Let $d\in\mathbb N$ and $M$ be a $d$-dimensional properly embedded $C^1$-submanifold of $\mathbb R^d$ with boundary. In the context of the trace operator, I've seen the usage of the $L^p$-space $L^1(\...
0xbadf00d's user avatar
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0 votes
1 answer
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Setting values of $ u$ in $H^{1/2}$ on a set of zero measure

everyone If I set values of $u \in L^2(\partial \Omega)$, $\Omega \subset R^3$, at several points on $\partial \Omega$, I will not not define $u$, because it is a set of zero measure. Am I correct ...
lumberman's user avatar
3 votes
2 answers
743 views

Different norms for the $H^{1/2}$ sobolev spaces

I am looking for references for the following results, which i believe to be true : Let $B$ a Lipschitz domain in $\mathbb{R}^d$, $f \in H^{1/2}(\partial B)$. We note $\gamma_0 : H^1(B) \mapsto H^{1/2}...
Velobos's user avatar
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1 vote
1 answer
151 views

Compactness of the trace operator in dimension 1

Let $T$ be the operator defined on the Sobolev space $H^1((0,1))$ by $$T:H^1((0,1)) \longrightarrow \mathbb{R} \\ f \mapsto f(0).$$ This operator is clearly a finite rank operator and thus it is ...
Gustave's user avatar
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