Well-Posed Problems for a Partial Differential Equation of Order $2m + 1$

Abstract

containing the elliptic differential operator M of order 2m and the differential operator L of order <__ 2m. Hilbert space methods are used to formulate and solve an abstract form of the problem and to discuss existence, uniqueness, asymptotic behavior and boundary conditions of a solution. The formulation of a generalized problem is the objective of 1, and we shall have reason to consider two types of solutions, called weak and strong. Sufficient conditions on the operatorM are given for the existence and uniqueness ofa weak solution to the generalized problem. These conditions constitute elliptic hypotheses onM and are discussed briefly in 3. Similar assumptions on L lead to results on the asymptotic behavior of a weak solution. The case in whichM and L are equal and self-adjoint is discussed in 2, and it is here that the role of the coefficient 7 of the equation appears first. Special as it is, this is a situation that often arises in applications, and there has been considerable interest in this coefficient 7 [4], [25]. The weak and strong solutions are distinguished not only by regularity conditions but also by their associated boundary conditions. It first appears in 5 that it is possible to prescribe too many (independent) boundary conditions on a strong solution, but in the applications it is seen that the interdependence of these conditions is built into the assumptions on the domains ofthe operators.Two examples of applications appear in 6 with a discussion of the types of boundary conditions that are appropriate. 1. The generalized problem. LetG be anonempty open set in the n-dimensional real Euclidean space, R, whose boundaryG is an (n 1)-dimensional manifold with G lying on one side of it. C(G) is the space ofinfinitely differentiable functions on G, and C(G) is the linear subspace of C(G) consisting of functions with compact support in G. The Sobolev space Hm(G)= H is the Hilbert space of (equivalence classes of) functions in L2(G), all of whose distributional derivatives through order m belong to L2(G). The inner product and norm are given,