A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic ...structures, among which are generalizations of Grassmann’s, Jacobi’s and Room’s identities. Moreover, such a view provides a higher dimensional analogue of the decomposition of the vector Laplacian, which itself gives an explicit index-free Helmholtz decomposition in arbitrary dimensions
In einer stärker computerisierten Welt finden Differential- und Differenzengleichungen immer mehr Anwendung. Das vorliegende Lehrbuch ist insbesondere für Studierende der ingenieurwissenschaftlichen, ...der informatikorientierten und der ökonomischen Studiengänge geeignet. Ausgewählte Kapitel sind auch für Schülerinnen und Schüler aus der Oberstufe mit den Leistungskursen Mathematik/Physik/Informatik interessant. Der präsentierte Stoff entspricht einer zweistündigen Vorlesung im Grundlagenbereich, wobei Basis-Kenntnisse aus der Analysis und der Linearen Algebra vorausgesetzt sind. Die Autoren zeigen Parallelen bei den Untersuchungen von linearen Differential- und linearen Differenzengleichungen auf, wobei die Vorgehensweisen anhand von vielen Beispielen ausführlich illustriert werden. Es werden lineare Differential- und lineare Differenzengleichungen erster und zweiter Ordnung betrachtet, sowie den Leserinnen und Leser alle Werkzeuge für die Betrachtungen von Gleichungen höherer Ordnung zur Verfügung gestellt.
In einer stärker computerisierten Welt finden Differential- und Differenzengleichungen immer mehr Anwendung. Das vorliegende Lehrbuch ist insbesondere für Studierende der ingenieurwissenschaftlichen, ...der informatikorientierten und der ökonomischen Studiengänge geeignet. Ausgewählte Kapitel sind auch für Schülerinnen und Schüler aus der Oberstufe mit den Leistungskursen Mathematik/Physik/Informatik interessant. Der präsentierte Stoff entspricht einer zweistündigen Vorlesung im Grundlagenbereich, wobei Basis-Kenntnisse aus der Analysis und der Linearen Algebra vorausgesetzt sind. Die Autoren zeigen Parallelen bei den Untersuchungen von linearen Differential- und linearen Differenzengleichungen auf, wobei die Vorgehensweisen anhand von vielen Beispielen ausführlich illustriert werden. Es werden lineare Differential- und lineare Differenzengleichungen erster und zweiter Ordnung betrachtet, sowie den Leserinnen und Leser alle Werkzeuge für die Betrachtungen von Gleichungen höherer Ordnung zur Verfügung gestellt.
For 1 <
p
<
∞
, we prove an
L
p
‐version of the generalized Korn inequality for incompatible tensor fields
P
in
. More precisely, let
be a bounded Lipschitz domain. Then there exists a constant
c
...=
c
(
p
, Ω) > 0 such that
holds for all tensor fields
, that is, for all
with vanishing tangential trace
on
∂
Ω where
ν
denotes the outward unit normal vector field to
∂
Ω. For compatible
, this recovers an
L
p
‐version of the classical Korn's first inequality and for skew‐symmetric
an
L
p
‐version of the Poincaré inequality.
For
$1< p<\infty$
we prove an
$L^{p}$
-version of the generalized trace-free Korn inequality for incompatible tensor fields
$P$
in
$W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
. ...More precisely, let
$\Omega \subset \mathbb {R}^{3}$
be a bounded Lipschitz domain. Then there exists a constant
$c>0$
such that
\ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \
holds for all tensor fields
$P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
, i.e., for all
$P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
with vanishing tangential trace
$P\times \nu =0$
on
$\partial \Omega$
where
$\nu$
denotes the outward unit normal vector field to
$\partial \Omega$
and
$\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$
denotes the deviatoric (trace-free) part of
$P$
. We also show the norm equivalence
\begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*}
for tensor fields
$P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$
. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset
$\Gamma \subseteq \partial \Omega$
of the boundary.
For $n\ge 2$ and $1 we prove an $L^p$-version of the generalized Korn-type inequality for incompatible, $p$-integrable tensor fields $P:\Omega \rightarrow \mathbb{R}^{n\,\times \,n}$ having ...$p$-integrable generalized $\underline{\operatorname{Curl}}\,$ and generalized vanishing tangential trace $P\,\tau _l=0$ on $\partial \Omega $, denoting by $\lbrace \tau _l\rbrace _{l=1,\,\ldots ,\,n-1}$ a moving tangent frame on $\partial \Omega $, more precisely we have: \ \left\Vert P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n\,\times \,n}\right)}\le c\,\left(\left\Vert \operatorname{sym}P \right\Vert _{L^p\left(\Omega ,\,\mathbb{R}^{n \times n}\right)}+ \left\Vert \underline{\operatorname{Curl}}\,P \right\Vert _{L^p\left(\Omega ,\,\left(\mathfrak{so}(n)\right)^n\right)}\right), \ where the generalized $\underline{\operatorname{Curl}}\,$ is given by $(\underline{\operatorname{Curl}}\,P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$ and $c=c(n,p,\Omega )>0$
For $1< p<\infty$ we prove an $L^{p}$-version of the generalized trace-free Korn inequality for incompatible tensor fields $P$ in $W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$. ...More precisely, let $\Omega \subset \mathbb {R}^{3}$ be a bounded Lipschitz domain. Then there exists a constant $c>0$ such that
\ \lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\leq c\,\left(\lVert{\operatorname{dev} \operatorname{sym} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \
holds for all tensor fields $P\in W^{1,p}_0(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$, i.e., for all $P\in W^{1,p} (\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$ with vanishing tangential trace $P\times \nu =0$ on $\partial \Omega$ where $\nu$ denotes the outward unit normal vector field to $\partial \Omega$ and $\operatorname {dev} P : = P -\frac 13 \operatorname {tr}(P) {\cdot } {\mathbb {1}}$ denotes the deviatoric (trace-free) part of $P$. We also show the norm equivalence
\begin{align*} &\lVert{ P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}+\lVert{ \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\\ &\quad\leq c\,\left(\lVert{P}\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})} + \lVert{ \operatorname{dev} \operatorname{Curl} P }\rVert_{L^{p}(\Omega,\mathbb{R}^{3\times 3})}\right) \end{align*}
for tensor fields $P\in W^{1,p}(\operatorname {Curl}; \Omega ,\mathbb {R}^{3\times 3})$. These estimates also hold true for tensor fields with vanishing tangential trace only on a relatively open (non-empty) subset $\Gamma \subseteq \partial \Omega$ of the boundary.
On $\alpha$-minimizing hypercones Lewintan, Peter
Rendiconti - Seminario matematico della Università di Padova,
01/2020, Letnik:
143
Journal Article
Recenzirano
In this paper we considerably extend the class of known $\alpha$-minimizing hypercones using sub-calibration methods. Indeed, the improvement of previous results follows from a careful analysis of ...special cubic and quartic polynomials.
For 1 < p < ∞, we prove an Lp‐version of the generalized Korn inequality for incompatible tensor fields P in
W01,p(Curl;Ω,ℝ3×3). More precisely, let
Ω⊂ℝ3 be a bounded Lipschitz domain. Then there ...exists a constant c = c(p, Ω) > 0 such that
‖P‖Lp(Ω,ℝ3×3)≤c‖symP‖Lp(Ω,ℝ3×3)+‖CurlP‖Lp(Ω,ℝ3×3)
holds for all tensor fields
P∈W01,p(Curl;Ω,ℝ3×3), that is, for all
P∈W1,p(Curl;Ω,ℝ3×3) with vanishing tangential trace
P×ν=0 on ∂Ω where ν denotes the outward unit normal vector field to ∂Ω. For compatible
P=Du, this recovers an Lp‐version of the classical Korn's first inequality and for skew‐symmetric
P=A an Lp‐version of the Poincaré inequality.
Abstract
For
$$n\ge 3$$
n
≥
3
and
$$1<p<\infty $$
1
<
p
<
∞
, we prove an
$$L^p$$
L
p
-version of the generalized trace-free Korn-type inequality for incompatible,
p
-integrable tensor fields
...$$P:\Omega \rightarrow \mathbb {R}^{n\times n}$$
P
:
Ω
→
R
n
×
n
having
p
-integrable generalized
$${\text {Curl}}_{n}$$
Curl
n
and generalized vanishing tangential trace
$$P\,\tau _l=0$$
P
τ
l
=
0
on
$$\partial \Omega $$
∂
Ω
, denoting by
$$\{\tau _l\}_{l=1,\ldots , n-1}$$
{
τ
l
}
l
=
1
,
…
,
n
-
1
a moving tangent frame on
$$\partial \Omega $$
∂
Ω
. More precisely, there exists a constant
$$c=c(n,p,\Omega )$$
c
=
c
(
n
,
p
,
Ω
)
such that
$$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$
‖
P
‖
L
p
(
Ω
,
R
n
×
n
)
≤
c
‖
dev
n
sym
P
‖
L
p
(
Ω
,
R
n
×
n
)
+
‖
Curl
n
P
‖
L
p
Ω
,
R
n
×
n
(
n
-
1
)
2
,
where the generalized
$${\text {Curl}}_{n}$$
Curl
n
is given by
$$({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$$
(
Curl
n
P
)
ijk
:
=
∂
i
P
kj
-
∂
j
P
ki
and "Equation missing"
denotes the deviatoric (trace-free) part of the square matrix
X
. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.
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