The Inverse Problem for Asgeirsson's Mean Value Equality

Abstract:

      For the linear partial differential operator with constant coefficients (?),the sufficient and necessary conditions for all C^m solutions u(x, y)of equation ( 4 ) satisfying Asgeirsson's mean value equality ( 2 ) are that L_x=c+a△_x where a(≠ 0 ), c are constants,△_x is the Laplacian.