Nonlinear Partial Differential Equations

Abstract

In this chapter we consider geometric partial differential equations, which appear for two-dimensional surfaces in their state of equilibrium. Here we give the differential-geometric foundations in Section 1 and determine in Section 2 the Euler equations of 2-dimensional, parametric functionals. In Section 3 we present the theory of characteristics for quasilinear hyperbolic differential equations, and Section 4 is devoted to the solution of Cauchy’s initial value problem with the aid of successive approximation. In Section 5 we treat the Riemannian integration method for linear hyperbolic differential equations. Finally, we prove S. Bernstein’s analyticity theorem in Section 6 using ideas of H. Lewy.