Optimal estimates for three general problems from the theory of elliptic differential equations will be studied. The first part of the project involves establishing an Agmon-Miranda maximum principle or showing the failure of such a principle for all constant coefficient linear elliptic equations in the dilation invariant setting of Lipschitz domain in Rn. Included here is the completion of the Lp theory in the manner of Agmon- Douglis-Nirenberg for Lipschitz domains. The second part involves strong maximum principles and best constant estimates for conditioned Brownian motion. The third part is concerned with optimal domain decomposition in the Schwarz iteration method employed in certain numerical problems.