Aissani: Thesis. University of Metz -in preparation ,
On a variational problem for an elastic membrane supporting a heavy ball, Cal. Var, vol.3, pp.447-473, 1995. ,
Variational inequalities and fluid flows in porous media, Applied Math Sciences Series fl52 Springer Verlag, 1984. ,
The contact set of a rigid body partially supported by a membrane, Nonlinear Analysis: Theory, Methods & Applications, vol.10, issue.3, pp.25-276, 1986. ,
DOI : 10.1016/0362-546X(86)90003-9
Stampacchia: An introduction to variational inequalities and their applications, 1980. ,
, , p.37, 1999.
,
, Chapitre 3 SLrr la défot-mation d'ttn frl éIastique pat' trois clisclrre^s... 1. Introduction Dans ce qui suit, on propose d'étudier la position d'équilibre de trois disques identiques rigides (Dt.,I < i < 3) roulant sur un fil élastique
, on se place dans un repère orthonormal dont I'axe des ordonnées est dirigé vers le bas. On représente par I'intervalle 0: (0, 1) la position initiale du fil éiastique, Comme dans le cas de deux disques identiques
, Chapitre 3. Sur la déformation d'un fr.I éIastique par trois disques
M: On the equilibrium position of a square on an elastic wire, Journal of Convex Analysis ,
S: On the deformation of an elastic wire by one or two heavy disks ,
M: Harmonic radius and concentration of energy, hyperbolic radius and Liouville's equation Au ,
On a variational problem for an elastic membrane supporting a heavy ball, Calculus of Variations and Partial Differential Equations, vol.20, issue.5, pp.447-473, 1995. ,
DOI : 10.1007/BF01187896
H: Problèmes unilatéraux, J. Math. Pures Appl, vol.51, pp.1-168 ,
D: The smoothness of solutions to nonlinear variational inequalities, Math. J, vol.23, pp.83-844 ,
A : A remark on the Hausdorff measure of a free boundary and the convergence of coincidence sets, Boll. U.M.I. 1S.A, issue.5, pp.109-113, 1981. ,
M: Variational inequalities and flow in porous media Applied Math Sciences series fl 52, 1984. ,
The contact set of a rigid body partially supported by a membrane, Nonlinear Analysis: Theory, Methods & Applications, vol.10, issue.3, pp.251-276, 1982. ,
DOI : 10.1016/0362-546X(86)90003-9
S: Elliptic partial differential equations of second order [13] Gustaffson. B: On the convexity of a solution of Liouvill's equation. Duke I\{ath Karvohl, B; When are solutions to nonlinear elliptic borrndary value problems convex?, J, vol.6014, issue.10, pp.303-311, 1985. ,
, How a minimal surface leaves an obstacle. Acta Math. 130' 22L-242, 1973. [16J Kinrlerlebrer. D -Siampacc]ria. G: An introdrrction to variatioual ineqrralitieç and their applications, 1980.
J: Cours de mathématiques, tome 1. Algèbre' Dunod, Paris, I97I. [18] Lions. J.L -Stampacchia. G: Variational inequalities, Comm. Pure Appl. Math, vol.20, pp.493-515, 1967. ,
Obstacle problems in mathematical physics. (North Holland Mathematics Studies) Amsterdam: I' ,
, W: CRC concise encyclopedia of mathematics, Assumptions Graph of lv, p.12, 1999.
, , 2001.
, Comparison between the two ca,ses (i) and (ii)
, Appendices 130
Interchanging the order of integration Let f ,g be continous functions respectively on IR+ x R+, R+. Then we have: Ir' Ir" f (u,s)s(s)itsilu = Io' {1,' ï{u,s)d,u}s(s)d,s ,
We can see that the integral, s)g(s)d,sd,uis taken over the triangle in Jo Jo figure al (o1) left-hand side of (al) gure a1 ,
, Therefore, vr'e can easily deduce the formula (al). l right-hand sicle of (al)
be continous and C bea nonnegative number If 7t f(ù<C+ | s(s)/(s)ds, 0(t<a, Jo then Proof' f t Suppose first that C > 0. Divide by C + Jo s@f?)ds and multiply by s(t) to obtain: 7t f (t)g(t)l{c * Jo gG)f G)ds} < g(t) ,
, An integration from 0 to t yields: 1tfi tog{(C+ / e(s)/(s)ds)lC}< | s(s)ds
, Jo Jo or ï(t) <c * [' gG)ïG)d,s 3C"xp{ / s(s)ds}
,
, take the limit as C + O throught positive values. This completes the proof. 1.t T(t) < Cexp{ | s(s)ds), pp.0-1
, JO Bibliography
K: Quantum dynamical semigroups and applications, Lecture notes in Physics, vol.286, 1987. ,
A: Volterra integral and differential equations Academic press, 1983. ,
C: Principles of differential and integral equations. Chelsea publishing company, p.97 ,
P: Methods of numerical integration. Cornputer science and applied Mathematics ,
G: The mathematicsof physics and chemistry, New Jersey. D Van Nostrand Company, INC: Maple V programming guide, 1956. ,
K: Virtues and limitations of Markovian master equations with a tim+dependerû generator, J Stat Phys, vol.100, p.314 ,
W: CRC concise Encyclopedia of Mathematics, 1999. ,