# 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.

90 questions
Filter by
Sorted by
Tagged with
1 vote
33 views

### 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 ...
• 47
1 vote
50 views

• 1,163
1 vote
56 views

### 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, $Tf = f|_{\partial \Omega}$ ...
• 6,009
32 views

### 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?
• 369
57 views

### 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{split} T\colon L^p(\Omega)& \rightarrow L^p(\partial\Omega)\\ u&\mapsto Tu=u_{|_{\partial \Omega}}...
• 369
1 vote
36 views

### 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....
1 vote
50 views

### 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 ...
• 123
1 vote
63 views

### 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, ...
• 914
104 views

### 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 ...
• 3,644
446 views

### 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 $g$, ...
• 1,482
1 vote
105 views

### 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 ...
• 2,771
1 vote
198 views

### 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 ...
• 2,109
49 views

### 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 ...
• 4,177
1 vote
126 views

### 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 ...
• 11
1k views

### 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}}(\...
21 views

### 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 ...
• 347
340 views

• 13.5k
### 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 ...