Boundedness of the Poisson-Dirichlet Problem on the Upper Half-Plane
The Dirichlet and Poisson problems are cornerstones of partial differential equations with vast applications in physics and have interested mathematicians for the last two centuries. In this study, we aim to connect the two problems and determine the boundedness of their solutions on the upper half-plane. For the Dirichlet problem, we derived an explicit solution as a convolution with the Poisson kernel. For the Poisson problem, we reformulated it as a linear superposition of an explicitly solvable Dirichlet-type problem and the free-space Poisson equation, the solution of which turns out to be a convolution with the fundamental solution. Using this result, we analyzed the Lp-boundedness of the Poisson solution in the upper half-plane by dominating its non-tangential norm by a Carleson-type measure. These results pave the way for future sharper estimates of these two problems in more complex domains.