Abstract
This chapter focuses on the qualitative properties of the heat and Laplace equation. The chapter reviews the weak maximum/minimum principle for the heat equation and for the Laplace equation. The Poisson integral formula in a circle and in a half-plane is reviewed. Also, the Gauss mean value theorem for harmonic functions in the plane and for harmonic functions in space are discussed. If u is harmonic in a region D (a connected bounded open region in two or three space dimensions) of the plane, the value of u (harmonic and continuous throughout D and on its boundary δD) at any interior point P of D is the average of the values of u around any circle centered on P and lying entirely inside D. While, if u is harmonic in a region D of space, the value of u at any interior point P of D is the average of the values of u around the surface of any sphere centered on P and lying entirely inside D.
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