The Dirichlet problem and the origin of Sobolev spaces

We will start this talk by introducing the classical Dirichlet problem. Investigating the origins of this problem in physics leads us to the classical Dirichlet principle on the minimizers of the energy functional. Sobolev spaces arise naturally as we try to find a suitable framework to study our problem. After reformulating our original problem in this new setting, functional analysis and the notion of weakly harmonic functions provide the missing link between the minimizers of the energy functional and solutions to the Dirichlet problem in the weak sense.