JUNCTION OF ONE-DIMENSIONAL MINIMIZATION PROBLEMS

Advances in Di↵erential Equations
Volume 13, Numbers 9-10 (2008), 935–958
JUNCTION OF ONE-DIMENSIONAL MINIMIZATION
PROBLEMS INVOLVING S 2 VALUED MAPS
Antonio Gaudiello
DAEIMI, Università degli Studi di Cassino
via G. Di Biasio 43, 03043 Cassino (FR), Italy
Rejeb Hadiji
Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées
CNRS UMR 8050, UFR Sciences et Technologie, 61
Avenue du Général de Gaulle Bat. P
4e étage, 94010 Créteil Cedex, France
(Submitted by: Jean-Michel Coron)
Abstract. This paper is composed of two parts. In the first part, via a
reduction dimension method, we derive a one-dimensional minimization
problem involving S 2 -valued maps for a thin T-shaped multidomain. In
the second one, we analyze this limit model.
1. Introduction
This paper, composed of two parts, carries on the research we started
in [9]. In the first part, via a reduction dimension method, we derive a
one-dimensional minimization problem involving S 2 -valued maps for a thin
T-shaped multidomain. In the second one, we analyze this limit model.
Let ⌦n ⇢ R3 , n 2 N, be a thin multidomain union of two joined orthogonal
cylinders: rn ⇥⇥[0, 1) and ( 12 , 12 )⇥rn (( 12 , 12 )⇥( 1, 0)), where (0, 0) 2 ⇥ ✓
( 12 , 21 ) ⇥ ( 12 , 12 ) and rn is a vanishing positive parameter (see Figure 1).
We point out that the first cylinder has constant height along the direction
x3 , the second one has constant height along the direction x1 , while both of
them have a small cross section and are joined by the surface {0} ⇥ rn ⇥.
For every n 2 N and 2 [0, +1), we consider the following minimization
problem:
nZ
En, := min
|DV (x1 , x2 , x3 )|2 d(x1 , x2 , x3 )
(1.1)
⌦n
Accepted for publication: June 2008.
AMS Subject Classifications: 78A25, 74K05, 74K30, 35B25.
935