bluuebunny/crossref_metadata_2025_split huggingface dataset:
Number of solutions with a norm bounded by a given constant of a semilinear elliptic PDE with a generic right-hand side
Authors: Nabutovsky Alexander 
Published: June 2012

We consider a semilinear boundary value problem

−
Δ
u
+
f
(
u
,
x
)
=
0

- \Delta u + f(u,x) = 0

in

Ω
⊂

R

N

\Omega \subset {\mathbb {R}^N}

and

u
=
0

u = 0

on

∂
Ω

\partial \Omega

. We assume that

f
f

is a

C
∞

{C^\infty }

-smooth function and

Ω
\Omega

is a bounded domain with a smooth boundary. For any

C
α

{C^\alpha }

-smooth perturbation

h
(
x
)

h(x)

of the right-hand side of the equation we consider the function

N
h

(
S
)

{N_h}(S)

defined as the number of

C

2
+
α

{C^{2 + \alpha }}

-smooth solutions

u
u

such that

‖
u
‖

C
0

(
Ω
)

≤
S

\left \| u\right \| _{{C^0}(\Omega )} \leq S

of the perturbed problem. How "small"

N
h

(
S
)

{N_h}(S)

can be made by a perturbation

h
(
x
)

h(x)

such that

‖
h
‖

C
0

(
Ω
)

≤
ε
?

\left \| h\right \| _{{C^0}(\Omega )} \leq \varepsilon ?

We present here an explicit upper bound in terms of

ε
\varepsilon

,

S
S

and
\[

max

|

u

|

≤
S
,
x
∈

Ω
¯

‖

D
u
i

f
(
u
,
x
)
‖

(
i
∈
{
0
,
1
,
2
}
)
.

\max \limits _{|u| \leq S,x \in \bar \Omega } \left \| D_u^i f(u,x)\right \| \quad (i \in \{ 0,1,2\} ).

\]
If

S
S

is fixed then

h
h

can be chosen by such a way that the upper bound persists under small in

C
0

{C^0}

-topology perturbations of

h
h

. We present an explicit lower bound for the radius of the ball of such admissible perturbations.
Rigidity of 𝑝-completed classifying spaces of alternating groups and classical groups over a finite field
Authors: Ishiguro Kenshi 
Published: June 2012

A

p
p

-adic rigid structure of the classifying spaces of certain finite groups

π
\pi

, including alternating groups

A
n

{A_n}

and finite classical groups, is shown in terms of the maps into the

p
p

-completed classifying spaces of compact Lie groups. The spaces

(
B
π

)
p
∧

(B\pi )_p^ \wedge

have no nontrivial retracts. As an application, it is shown that

(
B

A
n

)
p
∧

≃
(
B

Σ
n

)
p
∧

(B{A_n})_p^ \wedge \simeq (B{\Sigma _n})_p^ \wedge

if and only if

n
≢
0
,
1
,

mod

p

n\not \equiv 0,1,\;\bmod \,p

. It is also shown that

(
B
S
L
(
n
,

F

q

)

)
p
∧

≃
(
B
G
L
(
n
,

F

q

)

)
p
∧

(BSL(n,{\mathbb {F}_q}))_p^ \wedge \simeq (BGL(n,{\mathbb {F}_q}))_p^ \wedge

where

q
q

is a power of

p
p

if and only if

(
n
,
q
−
1
)
=
1

(n,q - 1) = 1

.
Subgroup conditions for groups acting freely on products of spheres
Authors: Silverman Judith H. 
Published: June 2012

Let

d
d

and

h
h

be integers such that either

d
≥
2

d \geq 2

and

h
=

2
d

−
1

h = {2^d} - 1

, or

d
=
4

d = 4

and

h
=
5

h = 5

. Suppose that the group

G

\mathcal {G}

contains an elementary-abelian

2
2

-subgroup

E
d

{E_d}

of rank

d
d

with an element

σ
\sigma

of order

h
h

in its normalizer. We show that if

G

\mathcal {G}

admits a free and

F

2

{{\mathbf {F}}_2}

-cohomologically trivial action on

(

S
n

)
d

{({S^n})^d}

, then some nontrivial power of

σ
\sigma

centralizes

E
d

{E_d}

. The cohomology ring

H

∗

(

E
d

;

F

2

)
≃

F

2

[

y
1

,
…
,

y
d

]

{H^{\ast } }({E_d};{{\mathbf {F}}_2}) \simeq {{\mathbf {F}}_2}[{y_1}, \ldots ,{y_d}]

is a module over the Steenrod algebra

A

(
2
)

\mathcal {A}(2)

. Let

θ
∈

F

2

[

y
1

,
…
,

y
d

]

\theta \in {{\mathbf {F}}_2}[{y_1}, \ldots ,{y_d}]

, and let

c
≥
d
−
2

c \geq d - 2

be an integer. We show that

θ
\theta

divides

S

q

2
i

(
θ
)

S{q^{{2^i}}}(\theta )

in the polynomial ring for

0
≤
i
≤
c
⇔
θ
=

τ

2

c
−
d
+
3

π

0 \leq i \leq c \Leftrightarrow \theta = {\tau ^{{2^{c - d + 3}}}}\pi

, where

τ
\tau

divides

S

q

2
i

(
τ
)

S{q^{{2^i}}}(\tau )

for

0
≤
i
≤
d
−
3

0 \leq i \leq d - 3

and

π
\pi

is a product of linear forms.
Fourier inequalities with nonradial weights
Authors: Carton-Lebrun C. 
Published: June 2012

Let

F

f
(
γ
)
=

∫

R

n

(

e

−
2
i
π
γ
∙
x

−
1
)
f
(
x
)

d
x
,
n
>
1

\mathcal {F}\;f(\gamma ) = {\smallint _{{\mathbb {R}^n}}}({e^{ - 2i\pi \gamma \bullet x}} - 1)f(x)\,dx,n > 1

, and

u
u

,

v
v

be nonnegative functions. Sufficient conditions are found in order that

‖

F

f
‖

q
,
u

≤
C

‖
f
‖

p
,
v

\left \| \mathcal {F}\;f\right \| _{q,u} \leq C\left \| f\right \| _{p,v}

for all

f
∈

L
v
p

(

R

n

)

f \in L_v^p({\mathbb {R}^n})

. Pointwise and norm approximations of

F

f

\mathcal {F}\;f

are derived. Similar results are obtained when

u
u

is replaced by a measure weight. In the case

v
(
x
)
=

|

x

|

n
(
p
−
1
)

v(x) = |x{|^{n(p - 1)}}

, a counterexample is given which shows that no Fourier inequality can hold for all

f
f

in

L

c
,
0

∞

L_{c,0}^\infty

. Spherical restriction theorems are established. Further conditions for the boundedness of

F

\mathcal {F}

are discussed.
Generation and propagation of interfaces in reaction-diffusion systems
Authors: Chen Xinfu 
Published: June 2012

This paper is concerned with the asymptotic behavior, as

ε
↘
0

\varepsilon \searrow 0

, of the solution

(

u
ε

,

v
ε

)

({u^\varepsilon },{v^\varepsilon })

of the second initial-boundary value problem of the reaction-diffusion system:
\[

{

u
t
ε

−
ε
Δ

u
ε

=

1
ε

f
(

u
ε

,

υ
ε

)
≡

1
ε

[

u
ε

(
1
−

u

ε
2

)
−

υ
ε

]
,

υ
t
ε

−
Δ

υ
ε

=

u
ε

−
γ

υ
ε

\left \{ {\begin {array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \Delta {u^\varepsilon } = \frac {1}{\varepsilon }f({u^\varepsilon },{\upsilon ^\varepsilon }) \equiv \frac {1}{\varepsilon }[{u^\varepsilon }(1 - {u^{\varepsilon 2}}) - {\upsilon ^\varepsilon }],} \hfill \\ {\upsilon _t^\varepsilon - \Delta {\upsilon ^\varepsilon } = {u^\varepsilon } - \gamma {\upsilon ^\varepsilon }} \hfill \\ \end {array} } \right .

\]
where

γ
>
0

\gamma > 0

is a constant. When

v
∈
(
−
2

3

/

9
,
2

3

/

9
)

v \in ( - 2\sqrt 3 /9,2\sqrt 3 /9)

,

f
f

is bistable in the sense that the ordinary differential equation

u
t

=
f
(
u
,
v
)

{u_t} = f(u,v)

has two stable solutions

u
=

h
−

(
v
)

u = {h_ - }(v)

and

u
=

h
+

(
v
)

u = {h_ + }(v)

and one unstable solution

u
=

h
0

(
v
)

u = {h_0}(v)

, where

h
−

(
v
)
,

h
0

(
v
)

{h_ - }(v), {h_0}(v)

, and

h
+

(
v
)

{h_ + }(v)

are the three solutions of the algebraic equation

f
(
u
,
v
)
=
0

f(u,v) = 0

. We show that, when the initial data of

v
v

is in the interval

(
−
2

3

/

9
,
2

3

/

9
)

( - 2\sqrt 3 /9,2\sqrt 3 /9)

, the solution

(

u
ε

,

v
ε

)

({u^\varepsilon },{v^\varepsilon })

of the system tends to a limit

(
u
,
v
)

(u,v)

which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function

u
u

is a "phase" function in the sense that it coincides with

h
+

(
v
)

{h_ + }(v)

in one region

Ω
+

{\Omega _ + }

and with

h
−

(
v
)

{h_ - }(v)

in another region

Ω
−

{\Omega _ - }

. The common boundary (free boundary or interface) of the two regions

Ω
−

{\Omega _ - }

and

Ω
+

{\Omega _ + }

moves with a normal velocity equal to

V

(
v
)

\mathcal {V}(v)

, where

V

(
∙
)

\mathcal {V}( \bullet )

is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially

u
(
∙
,
0
)
−

h
0

(
v
(
∙
,
0
)
)

u( \bullet , 0) - {h_0}(v( \bullet , 0))

takes both positive and negative values, then an interface will develop in a short time

O
(
ε

|

ln
⁡
ε

|

)

O(\varepsilon |\ln \varepsilon |)

near the hypersurface where

u
(
x
,
0
)
−

h
0

(
v
(
x
,
0
)
)
=
0

u(x,0) - {h_0}(v(x,0)) = 0

.
The cohomology algebra of a commutative group scheme
Authors: Fossum Robert , Haboush William 
Published: June 2012

Let

k
k

be a commutative ring with unit of characteristic

p
>
0

p > 0

and let

G
=
Spec
⁡
(
A
)

G = \operatorname {Spec}(A)

be an affine commutative group scheme over

k
k

. Let

H

∙

(
G
)

{{\text {H}}^ \bullet }(G)

be the graded Hochschild algebraic group cohomology algebra and, for

M
M

a rational

G
G

-module, let

H

∙

(
G
,
M
)

{{\text {H}}^ \bullet }(G,M)

denote the graded Hochschild cohomology

H

∙

(
G
)

{{\text {H}}^ \bullet }(G)

-module. We show that

H

∙

(
G
)

{{\text {H}}^ \bullet }(G)

is, in general, a graded Hopf algebra. When

G
=

G

a
,
k

G = {{\mathbf {G}}_{a,k}}

, let

α

p
ν

{\alpha _{{p^\nu }}}

denote the subgroup of

p
ν

{p^\nu }

-nilpotents and let

F

ν

{{\text {F}}_\nu }

denote the

ν
\nu

th power of the Frobenius. We show that for any finite

M
M

that there is a

ν
\nu

such that
\[

H

∙

(

G

a
,
k

,
M
)
≅

H

∙

(

α

p
ν

,
M
)

⊗
k

F

ν

∗

(

H

∙

(

G

a
,
k

)
)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}},M) \cong {{\text {H}}^ \bullet }({\alpha _{{p^\nu }}},M){ \otimes _k}{\text {F}}_{\nu }^\ast ({{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}}))

\]
where

F

ν
∗

{\text {F}}_\nu ^\ast

is the endomorphism of

H

∙

(

G

a
,
k

)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}})

induced by

F
v

{F_v}

. As a consequence, we can show that

H

∙

(

G

a
,
k

,
M
)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}},M)

is a finitely generated module over

H

∙

(

G

a
,
k

)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}})

when

M
M

is a finite dimensional vector space over

k
k

.
The constrained least gradient problem in 𝑅ⁿ
Authors: Sternberg Peter , Williams Graham , Ziemer William P. 
Published: June 2012

We consider the constrained least gradient problem
\[

inf

{

∫
Ω

|

∇
u

|

d
x
:
u
∈

C

0
,
1

(

Ω
¯

)
,

|

∇
u

|

≤
1

a.e.

,
u
=
g

on

∂
Ω

}

\inf \left \{ {\int _\Omega {|\nabla u|dx:u \in {C^{0,1}}(\bar \Omega ),\quad |\nabla u| \leq 1\;{\text {a.e.}},u = g\;{\text {on}}\;\partial \Omega } } \right \}

\]
which arises as the relaxation of a nonconvex problem in optimal design. We establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem. We also establish the uniqueness of solutions and discuss their structure.
Universal cover of Salvetti’s complex and topology of simplicial arrangements of hyperplanes
Authors: Paris Luis 
Published: June 2012

Let

V
V

be a real vector space. An arrangement of hyperplanes in

V
V

is a finite set

A

\mathcal {A}

of hyperplanes through the origin. A chamber of

A

\mathcal {A}

is a connected component of

V
−
(

∪

H
∈

A

H
)

V - ({ \cup _{H \in \mathcal {A}}}H)

. The arrangement

A

\mathcal {A}

is called simplicial if

∩

H
∈

A

H
=
{
0
}

{ \cap _{H \in \mathcal {A}}}H = \{ 0\}

and every chamber of

A

\mathcal {A}

is a simplicial cone. For an arrangement

A

\mathcal {A}

of hyperplanes in

V
V

, we set
\[

M
(

A

)
=

V

C

−

(

⋃

H
∈

A

H

C

)

,

M(\mathcal {A}) = {V_\mathbb {C}} - \left ({\bigcup \limits _{H \in \mathcal {A}} {{H_\mathbb {C}}} } \right ),

\]
where

V

C

=

C

⊗
V

{V_\mathbb {C}} = \mathbb {C} \otimes V

is the complexification of

V
V

, and, for

H
∈

A

H \in \mathcal {A}

,

H

C

{H_\mathbb {C}}

is the complex hyperplane of

V

C

{V_\mathbb {C}}

spanned by

H
H

. Let

A

\mathcal {A}

be an arrangement of hyperplanes of

V
V

. Salvetti constructed a simplicial complex

Sal
⁡
(

A

)

\operatorname {Sal}(\mathcal {A})

and proved that

Sal
⁡
(

A

)

\operatorname {Sal}(\mathcal {A})

has the same homotopy type as

M
(

A

)

M(\mathcal {A})

. In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex

Sal
^

(

A

)

\hat {\operatorname {Sal}}(\mathcal {A})

and prove that there is a natural map

p
:

Sal
^

(

A

)
→
Sal
⁡
(

A

)

p:\hat {\operatorname {Sal}}(\mathcal {A}) \to \operatorname {Sal}(\mathcal {A})

which is the universal cover of

Sal
⁡
(

A

)

\operatorname {Sal}(\mathcal {A})

. At the end, we use

Sal
^

(

A

)

\hat {\operatorname {Sal}}(\mathcal {A})

to give a new proof of Deligne’s result: "if

A

\mathcal {A}

is a simplicial arrangement of hyperplanes, then

M
(

A

)

M(\mathcal {A})

is a

K
(
π
,
1
)

K(\pi ,1)

space." Namely, we prove that

Sal
^

(

A

)

\hat {\operatorname {Sal}}(\mathcal {A})

is contractible if

A

\mathcal {A}

is a simplicial arrangement.
Vojta’s refinement of the subspace theorem
Authors: Schmidt Wolfgang M. 
Published: June 2012

Vojta’s refinement of the Subspace Theorem says that given linearly independent linear forms

L
1

,
…
,

L
n

{L_1}, \ldots , {L_n}

in

n
n

variables with algebraic coefficients, there is a finite union

U
U

of proper subspaces of

Q

n

{\mathbb {Q}^n}

, such that for any

ε
>
0

\varepsilon > 0

the points

x
_

_

∈

Z

n

∖
{

0
_

_

}

\underline {\underline x} \in {\mathbb {Z}^n}\backslash \{ \underline {\underline 0} \}

with (1)

|

L
1

(

x
_

_

)
⋯

L
n

(

x
_

_

)

|

>

|

x
_

_

|

−
ε

|{L_1}(\underline {\underline x} ) \cdots {L_n}(\underline {\underline x} )|\; > \;|\underline {\underline x} {|^{ - \varepsilon }}

lie in

U
U

, with finitely many exceptions which will depend on

ε
\varepsilon

. Put differently, if

X
(
ε
)

X(\varepsilon )

is the set of solutions of (1), if

X
¯

(
ε
)

\bar X(\varepsilon )

is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if

X
¯

′
(
ε
)

\bar X\prime (\varepsilon )

consists of components of dimension

>
1

> 1

, then

X
¯

′
(
ε
)
⊂
U

\bar X\prime (\varepsilon ) \subset U

. In the present paper it is shown that

X
¯

′
(
ε
)

\bar X\prime (\varepsilon )

is in fact constant when

ε
\varepsilon

lies outside a simply described finite set of rational numbers. More generally, let

k
k

be an algebraic number field and

S
S

finite set of absolute values of

k
k

containing the archimedean ones. For

υ
∈
S

\upsilon \in S

let

L
1
υ

,
…
,

L
m
υ

L_1^\upsilon , \ldots ,L_m^\upsilon

be linear forms with coefficients in

k
k

, and for

x
_

_

∈

K
n

∖
{

0
_

_

}

\underline {\underline x} \in {K^n}\backslash \{ \underline {\underline 0} \}

with height

H
k

(

x
_

_

)
>
1

{H_k}(\underline {\underline x} ) > 1

define

a

υ
i

(

x
_

_

)

{a_{\upsilon i}}(\underline {\underline x} )

by

|

L
i
υ

(

x
_

_

)

|

υ

/

|

x
_

_

|

υ

=

H
k

(

x
_

_

)

−

a

υ
i

(

x
_

_

)

/

d
υ

|L_i^\upsilon (\underline {\underline x} )|_\upsilon /|\underline {\underline x} |_\upsilon = {H_k}{(\underline {\underline x} )^{ - {a_{\upsilon i}}(\underline {\underline x} )/{d_\upsilon }}}

where the

d
υ

{d_\upsilon }

are the local degrees. The approximation set

A
A

consists of tuples

a
_

_

=
{

a

υ
i

}

(
υ
∈
S
,
1
≦
i
≦
m
)

\underline {\underline a} = \{ {a_{\upsilon i}}\} \;(\upsilon \in S,1 \leqq i \leqq m)

such that for every neighborhood

O
O

of

a
_

_

\underline {\underline a}

the points

x
_

_

\underline {\underline x}

with

{

a

v
i

{

x
_

_

)
}
∈
O

\{ {a_{{v_i}}}\{ \underline {\underline x} )\} \in O

are dense in the subspace topology. Then

A
A

is a polyhedron whose vertices are rational points.
Attracting and repelling point pairs for vector fields on manifolds. I
Authors: Meyer Gabriele 
Published: June 2012

Consider a compact, connected,

n
n

-dimensional, triangulable manifold

M
M

without boundary, embedded in

R

n
+
1

{{\mathbf {R}}^{n + 1}}

and a continuous vector field on

M
M

, given as a map

f
f

from

M
M

to

S
n

{S^n}

of degree not equal to

0
0

or

(
−
1

)

n
+
1

{( - 1)^{n + 1}}

. In this paper it is shown that there exists at least one pair of points

x
x

,

y
∈
M

y \in M

satisfying both

f
(
x
)
=
−
f
(
y
)

f(x) = - f(y)

and

f
(
x
)
=

x
−
y

|

|

x
−
y

|

|

f(x) = \frac {{x - y}} {{||x - y||}}

. Geometrically, this means, that the points and the vectors lie on one straight line and the vector field is "repelling". Similarly, if the degree of

f
f

is not equal to

0
0

or

1
1

, then there exists at least one "attracting" pair of points

x
x

,

y
∈
M

y \in M

satisfying both

f
(
x
)
=
−
f
(
y
)

f(x) = - f(y)

and

f
(
x
)
=

y
−
x

|

|

y
−
x

|

|

f(x) = \frac {{y - x}} {{||y - x||}}

. The total multiplicities are

k
∙
(
k
+

(
−
1
)

n

)

2

\frac {{k \bullet (k + {{( - 1)}^n})}} {2}

for repelling pairs and

k
∙
(
k
−
1
)

2

\frac {{k \bullet (k - 1)}} {2}

for attracting pairs. In the proof, we work with close simplicial approximations of the map

f
f

, using Simplicial, Singular and Čech Homology Theory, Künneth’s Theorem, Hopf’s Classification Theorem and the algebraic intersection number between two

n
n

-dimensional homology cycles in a

2
n

2n

-dimensional space. In the case of repelling pairs, we intersect the graph of

f
f

in

M
×

S
n

M \times {S^n}

with the set of points

(
x
,

x
−
y

|

|

x
−
y

|

|

)
∈
M
×

S
n

(x,\frac {{x - y}} {{||x - y||}}) \in M \times {S^n}

, where

x
x

and

y
y

satisfy that

f
(
x
)
=
−
f
(
y
)

f(x) = - f(y)

. In order to show that this set carries the homology

(
k
,
k
)
∈

H
n

(
M
×

S
n

,

Z

)

(k,k) \in {H_n}(M \times {S^n},{\mathbf {Z}})

, we study the set

A
f

≡
{
(
x
,
y
)
∈
M
×
M

|

f
(
x
)
=
−
f
(
y
)
}

{A_f} \equiv \{ (x,y) \in M \times M|f(x) = - f(y)\}

in a simplicial setting. Let

f
j

{f_j}

be a close simplicial approximation of

f
f

. It can be shown, that

A

f
j

{A_{{f_j}}}

is a homology cycle of dimension

n
n

with a natural triangulation and a natural orientation and that

A
f

{A_f}

and

A
f

j

{A_f}_j

carry the same homology.
The structure of the set of singular points of a codimension 1 differential system on a 5-manifold
Authors: Mormul P. , Zhitomirskiĭ M. Ya. 
Published: June 2012

Generic modules

V
V

of vector fields tangent to a 5-dimensional smooth manifold

M
M

, generated locally by four not necessarily linearly independent fields

X
1

{X_1}

,

X
2

{X_2}

,

X
3

{X_3}

,

X
4

{X_4}

, are considered. Denoting by

ω
\omega

the 1-form

{X_4}\lrcorner {X_3}\lrcorner {X_2}\lrcorner {X_1}\lrcorner \Omega \limits ^5 conjugated to

V
V

(

\Omega \limits ^5 is a fixed local volume form on

M
M

), the loci of singular behavior of

V
:

M

deg

(
V
)
=
{
p
∈
M

|

ω
(
p
)
=
0
}

V:{M_{\deg }}(V) = \{ p \in M|\omega (p) = 0\}

and

M

s
i
n
g

(
V
)
=
{
p
∈
M

|

ω
∧

(
d
ω

)
2

(
p
)
=
0
}

{M_{{\mathrm {sing}}}}(V) = \{ p \in M|\omega \wedge {(d\omega )^2}(p) = 0\}

are handled. The local classification of this pair of sets is carried out (outside a curve and a discrete set in

M

deg

(
V
)

{M_{\deg }}(V)

) up to a smooth diffeomorphism. In the most complicated case, around points of a codimension 3 submanifold of

M
M

,

M

s
i
n
g

(
V
)

{M_{{\mathrm {sing}}}}(V)

turns out to be diffeomorphic to the Cartesian product of

R

2

{\mathbb {R}^2}

and the Whitney’s umbrella in

R

3

{\mathbb {R}^3}

.
Radially symmetric solutions to a Dirichlet problem involving critical exponents
Authors: Castro Alfonso , Kurepa Alexandra 
Published: June 2012

In this paper we answer, for

N
=
3
,
4

N = 3,4

, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem

−
Δ
u
(
x
)
=
λ
u
(
x
)
+
u
(
x
)

|

u
(
x
)

|

4

/

(
N
−
2
)

- \Delta u(x) = \lambda u(x) + u(x)|u(x){|^{4/(N - 2)}}

,

x
∈
B
:=
{
x
∈

R
N

:

‖
x
‖

>
1
}

x \in B: = \{ x \in {R^N}:\left \| x \right \| > 1\}

,

u
(
x
)
=
0

u(x) = 0

,

x
∈
∂
B

x \in \partial B

, where

Δ
\Delta

is the Laplacean operator and

λ
>
0

\lambda > 0

. Indeed, we prove that if

N
=
3
,
4

N = 3,4

, then for any

λ
>
0

\lambda > 0

this problem has only finitely many radial solutions. For

N
=
3
,
4
,
5

N = 3,4,5

we show that, for each

λ
>
0

\lambda > 0

, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
On superquadratic elliptic systems
Authors: de Figueiredo Djairo G. , Felmer Patricio L. 
Published: June 2012

In this article we study the existence of solutions for the elliptic system
\[

−
Δ
u
=

∂
H

∂
v

(
u
,
v
,
x
)

in

Ω
,

−
Δ
v
=

∂
H

∂
u

(
u
,
v
,
x
)

in

Ω
,

u
=
0
,

v
=
0

on

∂
Ω
.

\begin {array}{*{20}{c}} { - \Delta u = \frac {{\partial H}}{{\partial v}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ { - \Delta v = \frac {{\partial H}}{{\partial u}}(u,v,x)\quad {\text {in}}\;\Omega ,} \\ {u = 0,\quad v = 0\quad {\text {on}}\;\partial \Omega .} \\ \end {array}

\]
where

Ω
\Omega

is a bounded open subset of

R

N

{\mathbb {R}^N}

with smooth boundary

∂
Ω

\partial \Omega

, and the function

H
:

R

2

×

Ω
¯

→

R

H:{\mathbb {R}^2} \times \bar \Omega \to \mathbb {R}

, is of class

C
1

{C^1}

. We assume the function H has a superquadratic behavior that includes a Hamiltonian of the form
\[

H
(
u
,
v
)
=

|

u

|

α

+

|

v

|

β

where

1
−

2
N

>

1
α

+

1
β

>
1

with

α
>
1
,
β
>
1.

H(u,v) = |u{|^\alpha } + |v{|^\beta }\quad {\text {where}}\;1 - \frac {2}{N} > \frac {1}{\alpha } + \frac {1}{\beta } > 1\;{\text {with}}\;\alpha > 1,\beta > 1.

\]
We obtain existence of nontrivial solutions using a variational approach through a version of the Generalized Mountain Pass Theorem. Existence of positive solutions is also discussed.
Weak solutions of hyperbolic-parabolic Volterra equations
Authors: Gripenberg Gustaf 
Published: June 2012

The existence of a global weak solution, satisfying certain a priori

L
∞

{L^\infty }

-bounds, of the equation

u
t

(
t
,
x
)
=

∫
0
t

k
(
t
−
s
)

(
σ
(

u
x

)

)
x

(
s
,
x
)
d
s
+
f
(
t
,
x
)

{u_t}(t,x) = \int _0^tk(t - s){(\sigma ({u_x}))_x}(s,x)ds + f(t,x)

is established. The kernel k is locally integrable and log-convex, and

σ
′

\sigma \prime

has only one local minimum which is positive.
On nonlinear delay differential equations
Authors: Iserles A. 
Published: June 2012

We examine qualitative behaviour of delay differential equations of the form
\[

y
′
(
t
)
=
h
(
y
(
t
)
,

y
(
q
t
)
)
,

y
(
0
)
=

y
0

,

y\prime (t) = h(y(t),\;y(qt)),\quad y(0) = {y_0},

\]
where h is a given function and

q
>
0

q > 0

. We commence by investigating existence of periodic solutions in the case of

h
(
u
,
v
)
=
f
(
u
)
+
p
(
v
)

h(u,v) = f(u) + p(v)

, where f is an analytic function and p a polynomial. In that case we prove that, unless q is a rational number of a fairly simple form, no nonconstant periodic solutions exist. In particular, in the special case when f is a linear function, we rule out periodicity except for the case when

q
=
1

/

deg
⁡
p

q = 1/\deg p

. If, in addition, p is a quadratic or a quartic, we show that this result is the best possible and that a nonconstant periodic solution exists for

q
=

1
2

q = \frac {1}{2}

or

1
4

\frac {1}{4}

, respectively. Provided that g is a bivariate polynomial, we investigate solutions of the delay differential equation by expanding them into Dirichlet series. Coefficients and arguments of these series are derived by means of a recurrence relation and their index set is isomorphic to a subset of planar graphs. Convergence results for these Dirichlet series rely heavily upon the derivation of generating functions of such graphs, counted with respect to certain recursively-defined functionals. We prove existence and convergence of Dirichlet series under different general conditions, thereby deducing much useful information about global behaviour of the solution.
Some cubic modular identities of Ramanujan
Authors: Borwein J. M. , Borwein P. B. , Garvan F. G. 
Published: June 2012

There is a beautiful cubic analogue of Jacobi’s fundamental theta function identity:

θ
3
4

=

θ
4
4

+

θ
2
4

\theta _3^4 = \theta _4^4 + \theta _2^4

. It is
\[

(

∑

n
,
m
=
−
∞

∞

q

n
2

+
n
m
+

m
2

)

3

=

(

∑

n
,
m
=
−
∞

∞

ω

n
−
m

q

n
2

+
n
m
+

m
2

)

3

+

(

∑

n
,
m
=
−
∞

∞

q

(
n
+

1
3

)

2

+
(
n
+

1
3

)
(
m
+

1
3

)
+

(
m
+

1
3

)

2

)

3

.

{\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{n^2} + nm + {m^2}}}} } \right )^3} = {\left ({\sum \limits _{n,m = - \infty }^\infty {{\omega ^{n - m}}{q^{{n^2} + nm + {m^2}}}} } \right )^3} + {\left ({\sum \limits _{n,m = - \infty }^\infty {{q^{{{(n + \frac {1}{3})}^2} + (n + \frac {1}{3})(m + \frac {1}{3}) + {{(m + \frac {1}{3})}^2}}}} } \right )^3}.

\]
Here

ω
=
exp
⁡
(
2
π
i

/

3
)

\omega = \exp (2\pi i/3)

. In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.