bluuebunny/crossref_metadata_2025 huggingface dataset:
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript f Baseline identical-to StartSet left-parenthesis x comma y right-parenthesis element-of upper M times upper M vertical-bar f left-parenthesis x right-parenthesis equals minus f left-parenthesis y right-parenthesis EndSet">

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.
A selection theorem for topological convex structures
Authors: van de Vel M.
Published: June 2012

A continuous selection theorem has been obtained for multivalued functions, the values of which are convex sets of certain synthetic convex structures. Applications are given related with superextensions, (semi)lattices, spaces of order arcs, trees, Whitney levels in hyperspaces, and geometric topology. Applications to traditional convexity in vector spaces involve Beer’s approximation theorem and a fixed point theorem of Dugundji-Granas. Some other applications (a.o. an invariant arc theorem) appear elsewhere.
3-manifolds which admit finite group actions
Authors: Wang Shi Cheng
Published: June 2012

We prove several results which support the following conjectures:

(
1
)

(1)

Any smooth action of a finite group on a geometric

3
3

-manifold can be conjugated to preserve the geometric structure.

(
2
)

(2)

Every irreducible closed

3
3

-manifold

M
M

with infinite

π
1

(
M
)

{\pi _1}(M)

is finitely covered by a Haken

3
3

-manifold.
L’espace des plongements d’un arc dans une surface
Authors: Cauty Robert
Published: June 2012

We prove that the space of embeddings of an arc into a surface without boundary

M
M

is homeomorphic to the product

U
(
M
)
×

l
2

U(M) \times {l^2}

, where

U
(
M
)

U(M)

is the unit tangent bundle of

M
M

.
Uniqueness of positive radial solutions of Δ𝑢+𝑓(𝑢)=0 in 𝑅ⁿ. II
Authors: McLeod Kevin
Published: June 2012

We prove a uniqueness result for the positive solution of

Δ
u
+
f
(
u
)
=
0

\Delta u + f(u) = 0

in

R

n

{\mathbb {R}^n}

which goes to

0
0

at

∞
\infty

. The result applies to a wide class of nonlinear functions

f
f

, including the important model case

f
(
u
)
=
−
u
+

u
p

f(u) = - u + {u^p}

,

1
>
p
>
(
n
+
2
)

/

(
n
−
2
)

1 > p > (n + 2)/(n - 2)

. The result is proved by reducing to an initial-boundary problem for the

ODE

u
+
(
n
−
1
)

/

r
+
f
(
u
)
=
0

{\text {ODE}}\;u + (n - 1)/r + f(u) = 0

and using a shooting method.
Applying coordinate products to the topological identification of normed spaces
Authors: Cauty Robert , Dobrowolski Tadeusz
Published: June 2012

Using the

l
2

{l^2}

-products we find pre-Hilbert spaces that are absorbing sets for all Borelian classes of order

α
≥
1

\alpha \geq 1

. We also show that the following spaces are homeomorphic to

Σ
∞

\Sigma ^\infty

, the countable product of the space

Σ
=
{
(

x
n

)
∈

R
∞

:
(

x
n

)

\Sigma = \{(x_n) \in R^\infty : (x_n)

is bounded}: (1) every coordinate product

∏
C

H
n

\prod _C H_n

of normed spaces

H
n

H_n

in the sense of a Banach space

C
C

, where each

H
n

H_n

is an absolute

F

σ
δ

F_{\sigma \delta }

-set and infinitely many of the

H
n

H_n

’s are

Z
σ

{Z_\sigma }

-spaces, (2) every function space

L
~

p

=

∩

p
′
>
p

L

p
′

\tilde {L}^p = \cap _{p\prime >p}L^{p\prime }

with the

L
q

{L^q}

-topology,

0
>
q
>
p
≤
∞

0>q>p \leq \infty

, (3) every sequence space

l
~

p

=

∩

p
>
p
′

l

p
′

{\tilde l^p} = { \cap _{p > p\prime }}{l^{p\prime }}

with the

l
q

l^q

-topology,

0
≤
p
>
q
>
∞

0 \leq p > q > \infty

. We also note that each additive and multiplicative Borelian class of order

α
≥
2

\alpha \geq 2

, each projective class, and the class of nonprojective spaces contain uncountably many topologically different pre-Hilbert spaces which are

Z
σ

Z_\sigma

-spaces.
Finite and countable additivity of topological properties in nice spaces
Authors: Tkachuk V. V.
Published: June 2012

Let

Q
∈

Q \in

character

≤
τ

\leq \tau

, pseudocharacter

≤
τ

\leq \tau

, tightness

≤
τ

\leq \tau

, weight

≤
τ

\leq \tau

,

P
τ

{P_\tau }

-property, discreteness, Fréchet-Urysohn property, sequentiality, radiality, pseudoradiality, local compactness,

k
k

-property. If

X
n

=
∪
{

X
i

:
i
∈
n
}

{X^n} = \cup \{ {X_i}:i \in n\}

,

X
i

⊢
Q

{X_i} \vdash Q

for all

i
∈
n

i \in n

then

X
⊢
Q

X \vdash Q

(i.e. the property

Q
Q

is

n
n

-additive in

X
n

{X^n}

for any

X
∈

T
3

X \in {T_3}

). Metrizability is

n
n

-additive in

X
n

{X^n}

provided

X
X

is compact or

c
(
X
)
=
ω

c(X) = \omega

.

ANR

{\text {ANR}}

-property is closely

n
n

-additive in

X
n

{X^n}

if

X
X

is compact ("closely" means additivity in case

X
i

{X_i}

is closed in

X
n

{X^n}

). If

Q
∈

Q \in

metrizability, character

≤
τ

\leq \tau

, pseudocharacter

≤
τ

\leq \tau

, diagonal number

≤
τ

\leq \tau

,

i
i

-weight

≤
τ

\leq \tau

, pseudoweight

≤
τ

\leq \tau

, local compactness then

Q
Q

is finitely additive in any topological group.
Orthogonal polynomials and hypergroups. II. The symmetric case
Authors: Lasser R.
Published: June 2012

The close relationship between orthogonal polynomial sequences and polynomial hypergroups is further studied in the case of even weight function, cf. [18]. Sufficient criteria for the recurrence relation of orthogonal polynomials are given such that a polynomial hypergroup structure is determined on

N

0

{\mathbb {N}_0}

. If the recurrence coefficients are convergent the dual spaces are determined explicitly. The polynomial hypergroup structure is revealed and investigated for associated ultraspherical polynomials, Pollaczek polynomials, associated Pollaczek polynomials, orthogonal polynomials with constant monic recursion formula and random walk polynomials.
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript normal s normal i normal n normal g Baseline left-parenthesis upper V right-parenthesis equals StartSet p element-of upper M vertical-bar omega logical-and left-parenthesis d omega right-parenthesis squared left-parenthesis p right-parenthesis equals 0 EndSet">

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}

.
Transference for radial multipliers and dimension free estimates
Authors: Auscher P. , Carro M. J.
Published: June 2012

For a large class of radial multipliers on

L
p

(

R

n

)

{L^p}({{\mathbf {R}}^{\mathbf {n}}})

, we obtain bounds that do not depend on the dimension n. These estimates apply to well-known multiplier operators and also give another proof of the boundedness of the Hardy-Littlewood maximal function over Euclidean balls on

L
p

(

R

n

)

{L^p}({{\mathbf {R}}^{\mathbf {n}}})

,

p
≥
2

p \geq 2

, with constant independent of the dimension. The proof is based on the corresponding result for the Riesz transforms and the method of rotations.
𝑝-adic hyperbolic planes and modular forms
Authors: Rhodes John A.
Published: June 2012

For

K
K

a number field and

p

{\mathbf {p}}

a finite prime of

K
K

, we define a

p

{\mathbf {p}}

-adic hyperbolic plane and study its geometry under the action of

G

L
2

(

K

p

)

G{L_2}({K_{\mathbf {p}}})

. Seeking an operator with properties analogous to those of the non-Euclidean Laplacian of the classical hyperbolic plane, we investigate the fundamental invariant integral operator, the Hecke operator

T

p

{T_{\mathbf {p}}}

. Letting

S
S

be a finite set of primes of a totally real

K
K

(including all the infinite ones), a modular group

Γ
(
S
)

\Gamma (S)

is defined. This group acts discontinuously on a product of classical and

p

{\mathbf {p}}

-adic hyperbolic planes.

S
S

-modular forms and their associated Dirichlet series are studied.
Product recurrence and distal points
Authors: Auslander J. , Furstenberg H.
Published: June 2012

Recurrence is studied in the context of actions of compact semigroups on compact spaces. (An important case is the action of the Stone-Čech compactification of an acting group.) If the semigroup E acts on the space X and F is a closed subsemigroup of E, then x in X is said to be F-recurrent if

p
x
=
x

px = x

for some

p
∈
F

p \in F

, and product F-recurrent if whenever y is an F-recurrent point (in some space Y on which E acts) the point (x, y) in the product system is F-recurrent. The main result is that, under certain conditions, a point is product F-recurrent if and only if it is a distal point.
A proof of 𝐶¹ stability conjecture for three-dimensional flows
Authors: Hu Sen
Published: June 2012

We give a proof of the

C
1

{C^1}

stability conjecture for three-dimensional flows, i.e., prove that there exists a hyperbolic structure over the

Ω
\Omega

set for the structurally stable three-dimensional flows. Mañé’s proof for the discrete case motivates our proof and we find his perturbation techniques crucial. In proving this conjecture we have overcome several new difficulties, e.g., the change of period after perturbation, the ergodic closing lemma for flows, the existence of dominated splitting over

Ω
∖

P

\Omega \backslash \mathcal {P}

where

P

\mathcal {P}

is the set of singularities for the flow, the discontinuity of the contracting rate function on singularities, etc. Based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems, i.e., we create unstable saddle connections if there are accumulations of periodic orbits on the singularities.
Transfer functions of regular linear systems. I. Characterizations of regularity
Authors: Weiss George
Published: June 2012

We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form

x
˙

(
t
)
=
A
x
(
t
)
+
B
u
(
t
)

\dot x(t) = Ax(t) + Bu(t)

,

y
(
t
)
=
C
x
(
t
)
+
D
u
(
t
)

y(t) = Cx(t) + Du(t)

, like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Fourés and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.
Classification of rank-2 ample and spanned vector bundles on surfaces whose zero loci consist of general points
Authors: Noma Atsushi
Published: June 2012

Let X be an n-dimensional smooth projective variety over an algebraically closed field k of characteristic zero, and E an ample and spanned vector bundle of rank n on X. To study the geometry of (X, E) in view of the zero loci of global sections of E, Ballico introduces a numerical invariant

s
(
E
)

s(E)

. The purposes of this paper are to give a cohomological interpretation of

s
(
E
)

s(E)

, and to classify ample and spanned rank-2 bundles E on smooth complex surfaces X with

s
(
E
)
=
2

c
2

(
E
)

s(E) = 2{c_2}(E)

, or

2

c
2

(
E
)
−
1

2{c_2}(E) - 1

; namely ample and spanned 2-bundles whose zero loci of global sections consist of general

c
2

(
E
)

{c_2}(E)

points or general

c
2

(
E
)
−
1

{c_2}(E) - 1

points plus one. As an application of these classification, we classify rank-2 ample and spanned vector bundles E on smooth complex projective surfaces with

c
2

(
E
)
=
2

{c_2}(E) = 2

.
Integer points on curves of genus two and their Jacobians
Authors: Grant David
Published: June 2012

Let C be a curve of genus 2 defined over a number field, and

Θ
\Theta

the image of C embedded into its Jacobian J. We show that the heights of points of J which are integral with respect to

[
2

]
∗

Θ

{[2]_\ast }\Theta

can be effectively bounded. As a result, if P is a point on C, and

P
¯

\bar P

its image under the hyperelliptic involution, then the heights of points on C which are integral with respect to P and

P
¯

\bar P

can be effectively bounded, in such a way that we can isolate the dependence on P, and show that if the height of P is bigger than some bound, then there are no points which are S-integral with respect to P and

P
¯

\bar P

. We relate points on C which are integral with respect to P to points on J which are integral with respect to

Θ
\Theta

, and discuss approaches toward bounding the heights of the latter.
Calculating discriminants by higher direct images
Authors: Weyman Jerzy
Published: June 2012

The author uses the homological algebra to construct for any line bundle

L

\mathcal {L}

on a nonsingular projective variety X the complex

F

(

L

)

\mathbb {F}(\mathcal {L})

whose determinant is equal to the equation of the dual variety

X

V

{X^{\text {V}}}

. This generalizes the Cayley-Koszul complexes defined by Gelfand, Kapranov and Zelevinski. The formulas for the codimension and degree of

X

V

{X^{\text {V}}}

in terms of complexes

F

(

L

)

\mathbb {F}(\mathcal {L})

are given. In the second part of the article the general technique is applied to classical discriminants and hyperdeterminants.
On the number of singularities in meromorphic foliations on compact complex surfaces
Authors: Ballico Edoardo
Published: June 2012

Here we study meromorphic foliations with singularities on a smooth compact complex surface. Aim: study the locations of the singularities, using vector bundle techniques and techniques introduced in the cohomological study of projective geometry.
Approximation from shift-invariant subspaces of 𝐿₂(𝐑^{𝐝})
Authors: de Boor Carl , DeVore Ronald A. , Ron Amos
Published: June 2012

A complete characterization is given of closed shift-invariant subspaces of

L
2

(

R

d

)

{L_2}({\mathbb {R}^d})

which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.
On the discriminant of a hyperelliptic curve
Authors: Lockhart P.
Published: June 2012

The minimal discriminant of a hyperelliptic curve is defined and used to generalize much of the arithmetic theory of elliptic curves. Over number fields this leads to a higher genus version of Szpiro’s Conjecture. Analytically, the discriminant is shown to be related to Siegel modular forms of higher degree.
Solutions to the quantum Yang-Baxter equation arising from pointed bialgebras
Authors: Radford David E.
Published: June 2012

Let

R
:
M
⊗
M
→
M
⊗
M

R:M \otimes M \to M \otimes M

be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. We introduce a quotient

A

red

(
R
)

{A^{{\text {red}}}}(R)

of the bialgebra

A
(
R
)

A(R)

constructed by Fadeev, Reshetihkin and Takhtajan, whose characteristics seem to more faithfully reflect properties R possesses as a linear operator. We characterize all R such that

A

red

(
R
)

{A^{{\text {red}}}}(R)

is a pointed bialgebra, and we determine all solutions R to the quantum Yang-Baxter equation when

A

red

(
R
)

{A^{{\text {red}}}}(R)

is pointed and

dim
⁡
M
=
2

\dim M = 2

(with a few technical exceptions when k has characteristic 2). Extensions of such solutions to the quantum plane are studied.
Quasiconformal mappings and Ahlfors-David curves
Authors: MacManus Paul
Published: June 2012

We show that if

ρ
\rho

is a quasiconformal mapping of the plane whose dilatation satisfies a certain quadratic Carleson measure condition relative to an Ahlfors-David curve

Γ
\Gamma

then

ρ
\rho

is differentiable almost everywhere on

Γ
\Gamma

and

log
⁡

|

ρ
′

|

∈

BMO

\log |\rho \prime | \in {\text {BMO}}

. When

Γ
\Gamma

is chord-arc we show that its image is a Bishop-Jones curve. If the Carleson norm is small then we show that

ρ
\rho

is absolutely continuous on

Γ
\Gamma

, the image of

Γ
\Gamma

is an Ahlfors-David curve, and

ρ
′
=

e
a

\rho \prime = {e^a}

, where

a
∈

BMO

a \in {\text {BMO}}

with a small norm.
Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d’isométries des arbres
Authors: Coornaert M. , Papadopoulos A.
Published: June 2012

Let X be a complete locally compact metric tree and

Γ
\Gamma

a group of isometries of X acting properly on this space. The space of bi-infinite geodesics in X constitutes a space GX on which

Γ
\Gamma

acts properly. Let

Ω
\Omega

be the quotient of GX by this action. The geodesic flow associated to

Γ
\Gamma

is the flow on

Ω
\Omega

which is the quotient of the geodesic flow on GX, defined by the time-shift on geodesics. To any

Γ
\Gamma

-conformal measure on the boundary

∂
X

\partial X

there is an associated measure m on

Ω
\Omega

which is invariant by the geodesic flow. We prove the following results: The geodesic flow on

(
Ω
,
m
)

(\Omega ,m)

is either conservative or dissipative. If it is conservative, then it is ergodic, If it is dissipative, then it is not ergodic unless it is measurably conjugate to the action of

R

\mathbb {R}

on itself by conjugation. We prove also a dichotomy in terms of the conical limit set

Λ
c

⊂
∂
X

{\Lambda _c} \subset \partial X

of

Γ
\Gamma

: the flow on

(
Ω
,
m
)

(\Omega ,m)

is conservative if and only if

μ
(

Λ
c

)
=
μ
(
∂
X
)

\mu ({\Lambda _c}) = \mu (\partial X)

, and it is dissipative if and only if

μ
(

Λ
c

)
=
0

\mu ({\Lambda _c}) = 0

. The results are analogous to results of E. Hopf and D. Sullivan in the case of Riemannian manifolds of constant negative curvature.
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta u left-parenthesis x right-parenthesis equals lamda u left-parenthesis x right-parenthesis plus u left-parenthesis x right-parenthesis StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript 4 slash left-parenthesis upper N minus 2 right-parenthesis">

−
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row minus normal upper Delta u equals StartFraction partial-differential upper H Over partial-differential v EndFraction left-parenthesis u comma v comma x right-parenthesis in normal upper Omega comma 2nd Row minus normal upper Delta v equals StartFraction partial-differential upper H Over partial-differential u EndFraction left-parenthesis u comma v comma x right-parenthesis in normal upper Omega comma 3rd Row u equals 0 comma v equals 0 on partial-differential normal upper Omega period EndLayout">

−
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis u comma v right-parenthesis equals StartAbsoluteValue u EndAbsoluteValue Superscript alpha Baseline plus StartAbsoluteValue v EndAbsoluteValue Superscript beta Baseline where 1 minus StartFraction 2 Over upper N EndFraction greater-than StartFraction 1 Over alpha EndFraction plus StartFraction 1 Over beta EndFraction greater-than 1 with alpha greater-than 1 comma beta greater-than 1 period">

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.
On the force between rotating co-axial black holes
Authors: Weinstein Gilbert
Published: June 2012

We study the force between rotating coaxial black holes, as it was defined in [9 and 10]. We show that under a certain limit, the force is attractive, and in fact tends to infinity. This lends support to the conjecture that the force is always positive.
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline left-parenthesis t comma x right-parenthesis equals integral Subscript 0 Superscript t Baseline k left-parenthesis t minus s right-parenthesis left-parenthesis sigma left-parenthesis u Subscript x Baseline right-parenthesis right-parenthesis Subscript x Baseline left-parenthesis s comma x right-parenthesis d s plus f left-parenthesis t comma x right-parenthesis">

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.
Escape rate for 2-dimensional Brownian motion conditioned to be transient with application to Zygmund functions
Authors: Housworth Elizabeth Ann
Published: June 2012

The escape rate of a 2-dimensional Brownian motion conditioned to be transient is determined to be

P
{
X
(
t
)
>
f
(
t
)

P\{ X(t) > f(t)

i.o. as

t
↑
∞
}
=
0

t \uparrow \infty \} = 0

or 1 according as

∑

n
=
1

∞

e

−
n

log
⁡
f
(

e

e
n

)
>

\sum \nolimits _{n = 1}^\infty {{e^{ - n}}\log f({e^{{e^n}}}) > }

or

=
∞

= \infty

. The result is used to construct a complex-valued Zygmund function—as a lacunary series—whose graph does not have

σ
\sigma

-finite linear Hausdorff measure. This contrasts the result of Mauldin and Williams that the graphs of all real-valued Zygmund functions have

σ
\sigma

-finite linear Hausdorff measure.
On nonlinear delay differential equations
Authors: Iserles A.
Published: June 2012

We examine qualitative behaviour of delay differential equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y prime left-parenthesis t right-parenthesis equals h left-parenthesis y left-parenthesis t right-parenthesis comma y left-parenthesis q t right-parenthesis right-parenthesis comma y left-parenthesis 0 right-parenthesis equals y 0 comma">

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.
Fibrations of classifying spaces
Authors: Ishiguro Kenshi , Notbohm Dietrich
Published: June 2012

We investigate fibrations of the form

Z
→
Y
→
X

Z \to Y \to X

, where two of the three spaces are classifying spaces of compact connected Lie groups. We obtain certain finiteness conditions on the third space which make it also a classifying space. Our results allow to express some of the basic notions in group theory in terms of homotopy theory, i.e., in terms of classifying spaces. As an application we prove that every retract of the classifying space of a compact connected Lie group is again a classifying space.
The Cauchy problem in 𝐶^{𝑁} for linear second order partial differential equations with data on a quadric surface
Authors: Johnsson Gunnar
Published: June 2012

By means of a method developed essentially by Leray some global existence results are obtained for the problem referred to in the title. The partial differential equations are required to have constant principal part and the initial surface to be irreducible and not everywhere characteristic. The Cauchy data are assumed to be given by entire functions. Under these conditions the location of all possible singularities of solutions are determined. The sets of singularities can be divided into two types, K- and L-singularities. K, the set of K-singularities, is the global version of the characteristic tangent defined by Leray. The L-sets are here quadric surfaces which, in contrast to the Ksets, allow unbounded singularities. The L-sets are in turn divided into three types: initial, asymptotic, and latent singularities. The initial singularities appear when the characteristic points of the initial surface are exceptional according to Leray’s local theory. These sets of singularity intersect the initial surface at characteristic points. The asymptotic case, where the set of singularities does not cut the initial surface, can be viewed as projectively equivalent to the initial case, the intersection taking place at infinite characteristic points. Finally the latent singularities are sets which intersect the initial surface, but where the solutions do not develop singularities initially. In the case of the Laplace equation with data on a real quadric surface it is shown that the K-singularities and the asymptotic singularities occur on the classical focal sets defined by Poncelet, Plücker, Darboux et al. There are also latent singularities appearing in coordinate subspaces of

R

N

{\mathbb {R}^N}

. As a corollary a new proof is given of the fact that ellipsoids have the Pompeiu property.
Semigroups and stability of nonautonomous differential equations in Banach spaces
Authors: Minh Nguyen Van
Published: June 2012

This paper is concerned with nonautonomous differential equations in Banach spaces. Using the theory of semigroups of linear and nonlinear operators one investigates the semigroups of weighted translation operators associated with the underlying equations. Necessary and sufficient conditions for different types of stability are given in terms of spectral properties of the translation operators and the differential operators associated with the equations.
Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis
Authors: Mordukhovich Boris
Published: June 2012

In this paper we develop a stability theory for broad classes of parametric generalized equations and variational inequalities in finite dimensions. These objects have a wide range of applications in optimization, nonlinear analysis, mathematical economics, etc. Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters. We employ a new approach of nonsmooth analysis based on the generalized differentiation of multivalued and nonsmooth operators. This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps. In particular, we prove new criteria for the existence of Lipschitzian multivalued and single-valued implicit functions.
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sigma-summation Underscript n comma m equals negative normal infinity Overscript normal infinity Endscripts q Superscript n squared plus n m plus m squared Baseline right-parenthesis cubed equals left-parenthesis sigma-summation Underscript n comma m equals negative normal infinity Overscript normal infinity Endscripts omega Superscript n minus m Baseline q Superscript n squared plus n m plus m squared Baseline right-parenthesis cubed plus left-parenthesis sigma-summation Underscript n comma m equals negative normal infinity Overscript normal infinity Endscripts q Superscript left-parenthesis n plus one third right-parenthesis squared plus left-parenthesis n plus one third right-parenthesis left-parenthesis m plus one third right-parenthesis plus left-parenthesis m plus one third right-parenthesis squared Baseline right-parenthesis cubed period">

(

∑

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.
The spectra of random pseudo-differential operators
Authors: Xia Jingbo
Published: June 2012

We study the spectra of random pseudo-differential operators generated by the same symbol function on different

L
2

{L^2}

-spaces. Our results generalize the spectral coincidence theorem of S. Kozlov and M. Shubin (Math. USSRSb. 51 (1985), 455-471) for elliptic operators of positive order associated with ergodic systems. Because of our new approach, we are able to treat operators of arbitrary order and associated with arbitrary dynamical systems. Furthermore, we characterize the spectra of these operators in terms of certain naturally obtained Borel measures on R.
Asymptotic measures for skew products of Bernoulli shifts with generalized north pole–south pole diffeomorphisms
Authors: Molinek D. K.
Published: June 2012

We study asymptotic measures for a certain class of dynamical systems. In particular, for

T
:

Σ
2

×
M
→

Σ
2

×
M

T:{\Sigma _2} \times M \to {\Sigma _2} \times M

, a skew product of the Bernoulli shift with a generalized north pole-south pole diffeomorphism, we describe the limits of the following two sequences of measures: (1) iterates under T of the product of Bernoulli measure with Lebesgue measure,

T
∗
n

(
μ
×
m
)

T_\ast ^n(\mu \times m)

, and (2) the averages of iterates of point mass measures,

1
n

Σ

k
=
0

n
−
1

δ

T
k

(
w
,
x
)

\frac {1}{n}\Sigma _{k = 0}^{n - 1}{\delta _{{T^k}(w,x)}}

. We give conditions for the limit of each sequence to exist. We also determine the subsequential limits in case the sequence does not converge. We exploit several properties of null recurrent Markov Chains and apply them to the symmetric random walk on the integers. We also make use of Strassen’s Theorem as an aid in determining subsequential limits.
Writing integers as sums of products
Authors: Chace Charles E.
Published: June 2012

In this paper we obtain an asymptotic expression for the number of ways of writing an integer N as a sum of k products of l factors, valid for

k
≥
3

k \geq 3

and

l
≥
2

l \geq 2

. The proof is an application of the Hardy-Littlewood method, and uses recent results from the divisor problem for arithmetic progressions.
Analysis and applications of holomorphic functions in higher dimensions
Authors: Yeh R. Z.
Published: June 2012

Holomorphic functions in

R
n

{R^n}

are defined to generalize those in

R
2

{R^2}

. A Taylor formula and a Cauchy integral formula are presented. An application of the Taylor formula to the kernel of the Cauchy integral formula results in Taylor series expansions of holomorphic functions. Real harmonic functions are expanded in series of homogeneous harmonic polynomials.
On the fundamental periods of Hilbert modular forms
Authors: Dou Ze-Li
Published: June 2012

The main purpose of this paper is to establish the existence of fundamental periods of primitive cusp forms of Hilbert modular type of several variables, as well as the relationship between those fundamental periods and the special values of the associated

L
L

-functions. These results, together with some recent results of Shimura, give us the means of translating with ease results concerning periods of automorphic forms derived from various points of view. We also verify several conjectures of Shimura on the properties of such fundamental periods.
Geometric consequences of extremal behavior in a theorem of Macaulay
Authors: Bigatti Anna , Geramita Anthony V. , Migliore Juan C.
Published: June 2012

F. S. Macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer-valued function which determine when such a function can be the Hilbert function of a standard graded

k
k

-algebra. We investigate some algebraic and geometric consequences which arise from the extremal cases of Macaulay’s theorem. Our work also builds on the fundamental work of G. Gotzmann. Our principal applications are to the study of Hilbert functions of zero-schemes with uniformity conditions. As a consequence, we have new strong limitations on the possible Hilbert functions of the points which arise as a general hyperplane section of an irreducible curve.
Generalization of the Whitney-Mahowald theorem
Authors: Li Bang He
Published: June 2012

The Whitney-Mahowald theorem gave normal Euler number

(
mod

4
)

(\bmod \,4)

for embeddings of a closed

2
n

2n

-manifold in Euclidean

4
n

4n

-space. We generalize this theorem to embeddings of closed

2
n

2n

-manifolds in an oriented

4
n

4n

-manifold with an approach in the framework of unoriented bordism groups of maps.
On orthogonal polynomials with respect to varying measures on the unit circle
Authors: Pan K.
Published: June 2012

Let

{

ϕ
n

(
d
μ
)
}

\{ {\phi _n}(d\mu )\}

be a system of orthonormal polynomials on the unit circle with respect to

d
μ

d\mu

and

{

ψ

n
,
m

(
d
μ
)
}

\{ {\psi _{n,m}}(d\mu )\}

be a system of orthonormal polynomials on the unit circle with respect to the varying measures

d
μ

/

|

w
n

(
z
)

|

2

,

z
=

e

i
θ

d\mu /|{w_n}(z){|^2},\;z = {e^{i\theta }}

, where

{

w
n

(
z
)
}

\{ {w_n}(z)\}

is a sequence of polynomials,

deg
⁡

w
n

=
n

\deg {w_n} = n

, whose zeros

w

n
,
1

,
…
,

w

n
,
n

{w_{n,1}}, \ldots ,{w_{n,n}}

lie in

|

z

|

>
1

|z| > 1

The asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained.
An elliptic regularity coefficient estimate for a problem arising from a frequency domain treatment of waves
Authors: Feng Xiaobing , Sheen Dongwoo
Published: June 2012

We consider a sequence of noncoercive elliptic problems, which are the wave equation in the frequency domain, in a rectangular or cubic domain with an absorbing boundary condition. The elliptic regularity coefficient depends on the frequency, and it has a singularity for both zero and infinite frequency. In this paper we derive an elliptic regularity estimate as the frequency tends to zero and infinity.
Nonproduct type analytic TUHF algebras
Authors: Ventura Belisario A.
Published: June 2012

We construct examples of nonproduct type real valued cocycles on a UHF groupoid, and show that the analytic triangular algebras associated to those cocycles, can only correspond to nonproduct type cocycles.
Groupoids associated with endomorphisms
Authors: Deaconu Valentin
Published: June 2012

To a compact Hausdorff space which covers itself, we associate an

r
r

-discrete locally compact Hausdorff groupoid. Its

C

∗

{{\mathbf {C}}^ * }

-algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the Cuntz algebra

O
p

{O_p}

. We consider examples related to coverings of the circle and of a Heisenberg

3
3

-manifold.
Analytic Fourier-Feynman transforms and convolution
Authors: Huffman Timothy , Park Chull , Skoug David
Published: June 2012

In this paper we develop an

L
p

{L_p}

Fourier-Feynman theory for a class of functionals on Wiener space of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x right-parenthesis equals f left-parenthesis integral Subscript 0 Superscript upper T Baseline alpha 1 d x comma ellipsis comma integral Subscript 0 Superscript upper T Baseline alpha Subscript n Baseline d x right-parenthesis">

F
(
x
)
=
f
(

∫
0
T

α
1

d
x
,
…
,

∫
0
T

α
n

d
x
)

F(x) = f(\int _0^T {{\alpha _1}dx, \ldots ,\int _0^T {{\alpha _n}dx)} }

. We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.
A discrete transform and Triebel-Lizorkin spaces on the bidisc
Authors: Wang Wei
Published: June 2012

We use a discrete transform to study the Triebel-Lizorkin spaces on bidisc

F
˙

p

α
q

,

f
˙

p

α
q

\dot F_p^{\alpha q},\,\dot f_p^{\alpha q}

and establishes the boundedness of transform

S
ϕ

:

F
˙

p

α
q

→

f
˙

p

α
q

{S_\phi }:\dot F_p^{\alpha q} \to \dot f_p^{\alpha q}

and

T
ψ

:

f
˙

p

α
q

→

F
˙

p

α
q

{T_\psi }:\dot f_p^{\alpha q} \to \dot F_p^{\alpha q}

. We also define the almost diagonal operator and prove its boundedness. With the use of discrete transform and Journé lemma, we get the atomic decomposition of

f
˙

p

α
q

\dot f_p^{\alpha q}

for

0
>
p
⩽
1
,

p
⩽
q
>
∞

0 > p \leqslant 1,\,p \leqslant q > \infty

. The atom supports on an open set, not a rectangle. Duality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis ModifyingAbove f With dot Subscript 1 Superscript alpha q Baseline right-parenthesis Superscript asterisk Baseline equals ModifyingAbove f With dot Subscript normal infinity Superscript minus alpha q Super Superscript prime Superscript Baseline comma StartFraction 1 Over q EndFraction plus StartFraction 1 Over q Superscript prime Baseline EndFraction equals 1 comma q greater-than 1 comma alpha element-of upper R">

(

f
˙

1

α
q

)

∗

=

f
˙

∞

−
α

q
′

,

1
q

+

1

q
′

=
1
,

q
>
1
,

α
∈
R

{(\dot f_1^{\alpha q})^{\ast }} = \dot f_\infty ^{ - \alpha q’},\,\tfrac {1} {q} + \tfrac {1} {{q’}} = 1,\,q > 1,\,\alpha \in R

, is established, too. The case for

F
˙

p

α
q

\dot F_p^{\alpha q}

is similar.
Hypersurfaces in space forms satisfying the condition Δ𝑥=𝐴𝑥+𝐵
Authors: Alías Luis J. , Ferrández Angel , Lucas Pascual
Published: June 2012

In this work we study and classify pseudo-Riemannian hypersurfaces in pseudo-Riemannian space forms which satisfy the condition

Δ
x
=
A
x
+
B

\Delta x = Ax + B

, where

A
A

is an endomorphism,

B
B

is a constant vector, and

x
x

stands for the isometric immersion. We prove that the family of such hypersurfaces consists of open pieces of minimal hypersurfaces, totally umbilical hypersurfaces, products of two nonflat totally umbilical submanifolds, and a special class of quadratic hypersurfaces.
The de Branges-Rovnyak model with finite-dimensional coefficients
Authors: Guyker James
Published: June 2012

A characterization in terms of the canonical model spaces of L. de Branges and J. Rovnyak is obtained for Hilbert spaces of formal power series with vector coefficients which satisfy a difference-quotient inequality, thereby extending the closed ideal theorems of A. Beurling and P. D. Lax.
Characterization of summability points of Nörlund methods
Authors: Grosse-Erdmann Karl-Goswin , Stadtmüller Karin
Published: June 2012

By a theorem of F. Leja any regular Nörlund method

(
N
,
p
)

(N,p)

sums a given power series

f
f

at most at countably many points outside its disc of convergence. This result was recently extended to a class of non-regular Nörlund methods by K. Stadtmüller. In this paper we obtain a more detailed picture showing how possible points of summability and the value of summation depend on

p
p

and

f
f

.
A free-boundary problem for the heat equation arising in flame propagation
Authors: Caffarelli Luis A. , Vázquez Juan L.
Published: June 2012

We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation energy. The problem asks for the determination of a domain in space-time,

Ω
⊂

R

n

×
(
0
,
T
)

\Omega \subset {{\mathbf {R}}^n} \times (0,T)

, and a function

u
(
x
,
t
)
⩾
0

u(x,t) \geqslant 0

defined in

Ω
\Omega

, such that

u
t

=
Δ
u

{u_t} = \Delta u

in

Ω
,

u

\Omega ,\;u

takes certain initial conditions,

u
(
x
,
0
)
=

u
0

(
x
)

u(x,0) = {u_0}(x)

for

x
∈

Ω
0

=
∂
Ω
∩
{
t
=
0
}

x \in {\Omega _0} = \partial \Omega \cap \{ t = 0\}

, and two conditions are satisfied on the free boundary

Γ
=
∂
Ω
∩
{
t
>
0
}
:
u
=
0

\Gamma = \partial \Omega \cap \{ t > 0\} :u = 0

and

u
ν

=
−
1

{u_\nu } = - 1

, where

u
ν

{u_\nu }

denotes the derivative of

u
u

along the spatial exterior normal to

Γ
\Gamma

. We approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data.