On a class of functions defined by partial differential equations

Abstract

LIPMAN BERS AND ABE GELBART(i) 1. Introduction.Various problems.inmechanics of continua lead to a system of differential equations of the form <riix)uz = Tiiy)vv, <r2ix)uy = -T2(y)»x, Consider the real part of (3.4), (3.5) f o-2udx -r2vdy.J cWe know that the partial derivatives ux, uy, vx exist.It follows from assumption ß that (ff2u)x, (<r2u)y, (r2v)x, (r2v)v also exist.By (1.1), ((r2u)y = (-t2v)x.This implies that (3.5) vanishes, by virtue of the theorem mentioned above.The same holds for the imaginary part of (3.4).From this proof it follows