Issue 42

J.-M. Nianga et alii, Frattura ed Integrità Strutturale, 42 (2017) 280-292; DOI: 10.3221/IGF-ESIS.42.30 286 Problem ( FVP  ): Find * * ( , ) u u inV V        such that: * * ( , ) ( , ) 0 ( , ) ( , ) 0 u a u v u b v u v V c d u V                               (46) In order to study the asymptotic behavior of the solution when  tends to zero, we use the classical following expansions, both for the unknown and the test functions: 0 1 2 2 ( ) ( ) ( , ) ( , ) ... u x u x u x y u x y        (47) 0 1 2 2 ( ) ( ) ( , ) ( , ) ... x x x y x y           (48) 0 1 2 2 ( ) ( ) ( , ) ( , ) ... v x v x v x y v x y        (49) 0 1 2 2 ( ) ( ) ( , ) ( , ) ... x x x y x y           (50) Introducing the following spaces:   1 ( ); ( ); u YC i i C V v v v H Y Y periodic     (51)   ( ); ; 0 u u YC i YC V v v v V v       (52)     * ( ); ; 0 u u YC i YC i i V v v v V v N on C     (53)   * * ( ); ; 0 u u YC i YC V v v v V v       (53)   1 ; ( ); YC C V H Y Y periodic       (54)   ; ; 0 YC YC V V           (55)     * ; ; 0 YC YC V V         (56)   * * ; ; 0 YC YC V V           (57) Comparing (47)-(50) with (46), we get the following relations: 0 0 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 ( ) ( ) ( ) ( ) ( ) ( ) j j j j j j k k k ijkl ijkl ijkl l i l i l i j j k i i i i ijkl ijk ijk l i k j k j v u v u v u u u u a dx a dx a dx x x x y y x v u u v u v u a dy e dx e y y x x x y                                                                             0 0 1 1 1 1 0 1 1 3 0 ( ) ( ) 0; , ( ( )) i i i i ijk ijk k j k j dx v u v u e dx e dy v v H y x y y                                                               (58)