In this chapter we begin solving PDEs in earnest. We will start with the homogeneous Dirichlet problem for the Laplacian, and introduce and develop theory for all of the necessary Sobolev spaces along the way. The Dirichlet problem will be presented in three equivalent formulations: as a distributional PDE, as a variational problem, and as a minimization problem. Inhomogeneous boundary conditions are introduced after a rigorous construction of the trace operator (restriction to the boundary) on Lipschitz domains in Chapter 4. We will consider the elliptic equation − Δ u = f in Ω , u = 0 on ∂ Ω , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath2_1.jpg"/> on an open subset Ω ⊂ R d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math2_1.jpg"/> . At this point, we will not make assumptions on the regularity of the boundary of the domain. It may be considered nonconvex, fractal, or poorly behaved in any other number of ways. The partial differential equation will be understood in the sense of distributions, i.e., that u and f are distributions, and that the equation holds when tested by an element of the set of test functions D ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math2_4.jpg"/> .