The Analytic Cauchy Problem for Fourth Order Elliptic Equations in Two Independent Variables

Abstract

An explicit representation in terms of the Riemann function is derived for the solution to the analytic Cauchy problem for a class of fourth order elliptic equations in two independent variables. Two particular cases are considered. For the biharmonic equation, results of an elementary form are obtained and compatibility conditions on the Cauchy data are found that guarantee regularity of the solution throughout a given domain. Representations in terms of axial data are found for solutions to the generalized axially symmetric biharmonic equation and to the iterated generalized axially symmetric Helmholtz equation. In principle, the derivation can be extended to higher order elliptic equations in two independent variables.