EQUATIONS OF BALANCE, HORIZONTAL DISTRIBUTIONS AND FOLIATION-VALUED SOLUTIONS

Abstract

Abstract Systems of second order partial differential equations of the form (equations of balance) are studied in a geometric context. The underlying space is K = G x IRnN, where G = Mn IRN is a space of fibers, with base manifold Mn for the independent variables and IRN for the range space of the dependent variables. A system of balance equations is shown to be generated by a single l-form W on K. All elements I (K) of A1(K) that generate identically satisfied equations of balance are characterized, I (K) being the identity class of l-forms on K. Regular mappings Y:G → K are studied. They are shown to induce completely integrable horizontal distributions on G. Balance systems are said to have foliation-valued solutions if every element of an n-dimensional foliation of G is the graph of a solution. A balance system is shown to have foliation-valued solutions if an element I of I(K) and a regular map Y:G → K can be found such that W—I ε ker(Y*) and Y induces a completely integrable horizontal distribution on G. Direct analogies with multiple integral problems in the calculus of variations are indicated.