 
Summary: Limit Problems for Plates Surrounded
by Soft Material
EMILIO ACERBI & GIUSEPPE BUTTAZZO
Communicated by E. GIUST!
I. In~oducfion
Consider an inhomogeneous clamped plate D, submitted to an external force
g(x). The (small) vertical displacement u(x) solves the minimum problem
min {/[.1 'E(x) }g~x) (l~ul2 2(1  cr(x)) detD2u) r g(x) u] dx: u EHI(D) ,
where E and a are the Young modulus and the Poisson coefficient respectively,
and D2u denotes the 2 · 2 matrix of second derivatives of u. We study a plate
having a central part/2 surrounded by an increasingly narrow annulus Z', made
of an increasingly soft material (i.e. the Young modulus E, tends to zero in X8)
The free energy of the plate is then
(1.1) F,(u) = _f l Eff~(]Aul 2  2(1  a) det Dzu)dx
//
+ 2  2Cl, ,)det , u dx
We study in particular the behavior as e> 0 of the solution u~ of
(1.2) min {F~(u)§ fg(x) udx:uEH~(~kJSe)}.
QV~ e
If re is the width of Se, we may have different limit problems depending on the
