The current monograph is meant to supply a accomplished and obtainable advent to the optimization of elliptic structures. This quarter of mathematical examine, which has many very important functions in technology and expertise. has skilled a magnificent improvement in the past twenty years. There are already many stable textbooks facing quite a few points of optimum layout difficulties. during this regard, we consult with the works of Pironneau , Haslinger and Neittaanmaki , , Sokolowski and Zolksio , Litvinov , Allaire , Mohammadi and Pironneau , Delfour and Zolksio , and Makinen and Haslinger . Already Lions [I9681 committed a tremendous a part of his classical monograph at the optimum regulate of partial differential equations to the optimization of elliptic platforms. allow us to additionally point out that even the first actual recognized challenge of the calculus of diversifications, the brachistochrone studied via Bernoulli again in 1696. is in reality a form optimization challenge. The usual richness of this mathematical learn topic, in addition to the super huge box of attainable functions, has created the bizarre state of affairs that even though many very important effects and techniques have already been est- lished, there are nonetheless urgent unsolved questions. during this monograph, we objective to deal with a few of these open difficulties; accordingly, there's just a minor overlap with the textbooks already latest within the box.