Linear Partial Differential Equations in ℝⁿ

Abstract

In this chapter we become familiar with the different types of partial differential equations in ℝⁿ. We treat the maximum principle for elliptic differential equations and prove the uniqueness of the mixed boundary value problem for quasilinear elliptic differential equations. Then we consider the initial value problem of the parabolic heat equation. Finally, we solve the Cauchy initial value problem for the hyperbolic wave equation in ℝⁿ and show its invariance under Lorentz transformations. The differential equations presented are situated in the center of mathematical physics.