The dissertation is made of two chapters. The first chapter is dedicated to the investigation of some properties of the layer potentials of a constant coefficient elliptic partial differential operator. In the second chapter, we focus our attention to the Lamè equations, which are related to the physic of an isotropic homogeneous elastic body. In particular, in the first chapter, we investigate the dependence of the single layer potential upon perturbation of the density, the support and the coefficients of the corresponding operator. Under some more restrictive assumptions on the operator, we prove a real analyticity theorem for the single layer potential and its derivatives. As a first step, we introduce a particular fundamental solution of a given constant coefficient partial differential operator. For this purpose, we exploite the construction of a fundamental solution given by John (1955). We have verified that, if the coefficients of the operator are constrained to a bounded set, then there exist a particular fundamental solution which is a sum of functions which depend real analytically on the coefficients of the operator. Such a result resembles the results of Mantlik (1991, 1992) (see also Tréves (1962)), where more general assumptions on the operator are considered. We observe that it is not a corollary. Indeed, we need a suitably detailed expression for the fundamental solution, which cannot be deduced by Mantlik's results. The next step is to introduce the support of our single layer potentials. It will be a compact sub-manifold of the the n-dimensional euclidean space parametrized by a suitable diffeomorphism defined on the boundary of a fixed domain. Then, we will be ready to state in Theorem 1.7 the main result of this chapter, which is a real analyticity result in the frame of Shauder spaces. The main idea of the proof stems from the papers of Lanza de Cristoforis & Preciso (1999) and by Lanza de Cristoforis & Rossi (2004, 2005) and exploits the Implicit Mapping Theorem for real analytic functions. Indeed, our main Theorem 1.7 is in some sense a natural extension of theorems obtained by Lanza de Cristoforis & Preciso (1999) and by Lanza de Cristoforis & Rossi (2004, 2005), for the Cauchy integral and for the Laplace and Helmholtz operators, respectively. Here we confine our attention to elliptic operators which can be factorized with operators of order 2. In the last section of the first chapter, we consider some applications of Theorem 1.7. In particular, we deduce a real analyticity theorem for the single and double layer potential which arise in the analysis of the boundary value problems for the Lamè equations and for the Stokes system. In the second chapter, we focus our attention to the Lamè equations. We consider some boundary value problems defined in a domain with a small hole. For each of them, we investigate the behavior of the solution and of the corresponding energy integral as the hole shrinks to a point. This kind of problem is not new at all and has been long investigated by the techniques of asymptotic analysis. It is perhaps difficult to give a complete list of contributions. Here we mention the work of Keller, Kozlov, Movchan, Maz'ya, Nazarov, Plamenewskii, Ozawa and Ward. The results that we present are in accordance with the behavior one would expect by looking at the above mentioned literature, but we adopt a different approach proposed by Lanza de Cristoforis (2001, 2002, 2005, 2007.) To do so, we exploit the real analyticity results for the elastic layer potentials obtained in the first chapter. We now briefly outline the main difference between our approach and the one of asymptotic analysis. Let d>0 be a parameter which is proportional …