The Dirichlet Problem for a Hyperbolic Equation

Abstract

This paper deals with the Dirichlet problem for the hyperbolic differential equation

$${U_{xy}} = 0$$

((1))

i. e. with the problem of determining a solution u of (1) from given values on a closed curve C.¹ Simple examples show that the Dirichlet problem for a hyperbolic equation has a completely different character from that of the corresponding problem for an elliptic equation such as the potential equation

$${U_{xx}} + {U_{yy}} = 0$$

((2))