In this chapter, we introduce the so-called Direct Method in the setting of Sobolev spaces, in order to get existence of minimizers for a functional with a prescribed boundary datum. Here as well, rather than considering the most general class of functionals, we will stick to a specific class, which is however general enough to encompass many important problems. As an application of the theory exposed, this chapter contains a thorough discussion on existence of solutions for some important partial differential equations coming from minimization problems, like: the torsional rigidity equation, the Lane-Emden equation and the eigenvalue equation for the Dirichlet-Laplacian. We also prove the Spectral Theorem for the Dirichlet-Laplacian on an open bounded set.