A priori estimates of the solution for the Dirichlet problem

We study the Dirichlet problem defined by Δu = f in Ω and u = g in δΩ when Ω is the half‐space or the unitary rectangle, obtaining an a priori estimate of the solution. Furthermore, in both cases a concrete numerical estimation is arrived at. First we get the a priori estimate in the case of the half‐space. The problem is resolved for the rectangle by initially translating it into the half‐space and using the results we had obtained for it, so that the problem can then be reduced back to the rectangle. Because of this we first establish an extension of the Sobolev space of second order in the rectangle to the one defined in the half‐space.