These lectures were presented at the joint C.I.M./CI.M.E. Summer School on
Optimal Shape Design held in Troia (Portugal), June 02 to June 07,1998. The
mathematical problems that can be described by the label "Optimal shape
design" form a broad area: it concerns the optimization of some performance
criterion where the criterion depends, besides constraints that qualify the
problem, on the "shape" of some region. A classical setting is Structural
Mechanics of elastic bodies such as bridges, beams, plates, shells, arches. These
structures have to satisfy requirements of load and have to be designed in an
optimal way; for example, should be built using the least amount of
material. Alternatively, one might seek the optimal shape of a geometrical object
moving in a fluid: B. Kawohl devotes most of his lectures to the classical
Newton's problem of minimal resistance. This fascinating problem, studied
by Newton in the interest of "Her Majesty's Navy", as reported by Kawohl,
goes back about three hundred years and has been a subject of controversy
and discussion from the very beginning (the functional to be minimized,
although rotationally symmetric, is not convex, and this explains the difficulty
of the problem). Alternatively, we may think of seeking the optimal shape of
a wing, to be designed so as to reduce the drag while keeping a given value
for the lift. Or we might wish to design the optimal shape of a region (a
harbor), given suitable constraints on the size of the entrance to the harbor,
subject to incoming waves, so as to minimize the height of the waves inside
it. Or we might wish to design some electrical device consisting of a (simply
connected) region (partially) coated with a conducting material, say copper
(the non-covered portion of the region is considered to be a perfect
insulator): the goal is to minimize the cost of the device, subject to constraints on
the performance of the resulting design. Or we might try to design materials
obtained layering several materials, with different characteristics: the goal in
this case could become that of computing the effective properties of the limit
material.