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Questions tagged [greens-relations]

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Identity involving fundamental solution of Laplace equation

I have the following identity If $\Omega$ is a smooth bounded domain and $u\in C^2(\bar\Omega)$ then for $x\in \Omega$ $$u(x)=\int\limits_\Omega \Phi(x-y)\Delta u(y)dy+\int\limits_{\partial\Omega}\...
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114 views

Green's identity and gradient estimate

After the proof of the Green's identity in the book "Han Q., Lin F. - Elliptic partial differential equations - AMS (1997)", they state at page 9: We may employ the local version of the Green's ...
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1answer
35 views

show that $u(x) = \frac{1}{2\pi} \int_{R^2} log(|y-x|)\Delta u(y)dy$

Let $u:R^2 \rightarrow R$ be a $C^2$ function with a compact support. I want to show that $u(x) = \frac{1}{2\pi} \int_{R^2} log(|y-x|)\Delta u(y)dy$, when $x, y \in R^2$. This look like it should be ...
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1answer
27 views

Green relations in semigroups : how to interpret $J(x) \setminus J_x$

I read in semigroup theory that given a semigroup $S^1$ (which has an identity), the $\mathcal{J}$ Green relation has an associated function $J(x)$: $$ J(x) = S^1xS^1 $$ which is the principal ideal ...
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1answer
40 views

How to apply Green formula?

I want to apply the Green formula two times in this integral: $$\int_0^{+\infty} (r(t) u^{(3)}(t))' v(t) dt $$ such that $u(0)=u'(0)=0$, $v(0)=v'(0)=0, u(+\infty)=u'(+\infty)=0, v(+\infty)=v'(+\infty)...
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1answer
55 views

Equal nr of $\mathscr D$-classes, different nr of idempotents

Are there examples of (finite) semigroups $S$ and $T$ such that they have the same 'number' of $\mathscr D$-classes, $S$ has idempotents and $T$ doesn't? Alternatively, they both may have idempotents, ...
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1answer
105 views

Expressing Green's relations in regular semigroups

Let $S$ be a semigroup and $a \in S$. An element $a' \in S$ is called an inverse of $a$ if $$ aa'a = a \qquad a'aa' = a'. $$ Denote the set of all inverses by $V(a)$. A semigroup where every ...
3
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1answer
53 views

Question on proof related to inclusion relations of principal ideals generated by elements.

Let $S$ be a semigroup, then an ideal $W$ is a subset $W \subseteq S$ such that $sWt \in W$ for all $s,t \in S^{\bullet}$, where $S^{\bullet}$ is $S$ with an additional unit element added if $S$ does ...
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1answer
101 views

$\mathcal{J}-$ trivial elements

What does it mean by saying that … idempotents are $\mathcal{J}-$ trivial. Indeed, by searching old docs and this one, we see that: A semigroup $S$ is $\mathcal{J}-$ trivial if two elements of ...
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Properties of Green's Equivalence $\mathcal L , \mathcal R, \mathcal D$ and $\mathcal H$ on a Semigroup

If $ar \mathcal R a$ , then the map $x \mapsto xr$ is a bijection from $\mathcal H_a$ onto $\mathcal H_{ar}$. I know that If D is an arbitrary $\mathcal D - $ class in a Semigroup $S$ and if $a, ...