The main focus of this study is to give new, constructive proofs to Korn's inequalities of every kind by using the properties of the divergence operator. Various proofs have been given to Korn's inequalities before, but the dependence on the domain of the constant appearing in these inequalities is not yet well known, save in some particular situations.

Estimates that quantify this dependence have applications in solid and fluid mechanics, when the domain under consideration depends on a scalar parameter. In this case, the magnitude of Korn's constant with respect to this parameter is essential in justifying dimensionally reduced models by means of convergence theorems when the parameter go to zero or go to infinity. Using a new approach that has the advantage of yielding constants that depend explicitly on several parameters associated with the domain, we give new proofs to Korn's inequalities of various kinds, and estimate at the same time the constants appearing in them.

We begin by showing how a particular inequality of Korn's type can be deduced from the surjectivity of the divergence operator between specific function spaces. Then we show how the other inequalities of Korn's type can be deduced from this particular inequality. We will consider several types of domains, typical in solid and fluid mechanics, for which we obtain sharp estimates of the corresponding Korn's inequalities.

Finally, we study several applications of these estimates in shell theory. We establish several inequalities of Korn's type in curvilinear coordinates on a shell, first on the original domain depending on the thickness of the shell, then on a fixed domain, independent of the thickness of the shell, which is the key to the asymptotic analysis of elastic shells leading to dimensionally reduced models of shells.