Description of the algebraic structure of second-order differential operators that can be represented in divergence form

Abstract

In the paper one proves that the second-order differential operators ℱ(x,u,u x ,u xx ), representable in divergence form, can be written as ℱ=cAΔA, where ΔA is the corresponding minor of order m of the Hessian $$\det (u_{xx} ) = \Delta _{\left( {\begin{array}{*{20}c} {1...n} \\ {1...n} \\ \end{array} } \right)} $$ whose representation coefficients cA=cA(x,u,u x ) satisfy certain algebraic relations. One introduces the concept of a second-order D-elliptic differential operator and one establishes that the totally elliptic operators of divergence form are linear with respect to the second derivatives of the function u.