(a) In 1890 Poincare [Po1] gave the

(a) In 1890 Poincare [Po1] gave the first complete proof, in rather general domains, of the existence and uniqueness of a solution of the Laplace equation for any continuous Dirichlet boundary condition. He introduced the so-called balayage method; this iterative method relies on solving the Dirichlet problem on balls in the domain and makes extensive use of the maximum principle and Harnack's inequality for harmonic functions. A systematic exposition of this method was given in his lectures of 1894-95 at the Sorbonne and published in [Po4]. Together with books of Harnack and Korn this is the origin of the extensive development of potential theory in the following decades. The interested reader will find a detailed summary of potential theory up to 1918 in the Encyklopadia article [Li2] of Lichtenstein. We note that, as pointed out in Section 19, the maximum principle for second order elliptic and parabolic equations has played a central role throughout the 20th century.