A Degree Theory for Lagrangian Boundary Value Problems

Abstract

Distribution theory steams from weak solutions of linear differential equations and it is hardly efficient for nonlinear equations. The use of distributions is actually difficult in linear boundary value problems, for no canonical duality theory is available for manifolds with boundary X . The scale of Sobolev-Slobodetskij spaces W (X ) makes it possible to consider the restrictions of functions to the boundary surface, however, these latter are defined only if s − 1/p > 0. To go beyond this range, one applies integral equalities obtained by manipulation of the Green formula. The study of general boundary value problems for differential equations in Sobolev-Slobodetskij spaces of negative smoothness goes back at least as far as [22]. For a boundary value problem, the Green formula is determined uniquely up to the counterpart of boundary data within the entire Cauchy data, see [26, 9.2.2]. This allows one to avoid much ambiguity in the choice of formal adjoint boundary value problem and to set up duality. As a result one is in a position to introduce weak solutions of the boundary value problem, see for instance Section 9.3.1 ibid. and elsewhere. The Cauchy data of a weak solution to an overdetermined elliptic system in the interior of X are proved to possess weak boundary values at ∂X if and only if the solution is of finite order of growth near the boundary surface, see [26, 9.3.6]. When considering a boundary value problem for a nonlinear equation, one has no good guide to an appropriate concept of weak solution. Perhaps one has to pass to the linearised problem. In any case the definition of a weak solution is implicitly contained in the variational setting of the boundary value problem. If the problem itself fails to be Lagrangian, it can be relaxed to variational one. It is just the task of experienced researcher to recover the concept of weak solution in the variational formulation, see [2].