Lagrange Multipliers, Morses Indices and Compactness

Abstract

The purpose of this paper is to emphasize two observations on variational problems. In some sense, they are independent even if they can be combined in some problems. The first one concerns the so-called strict subadditivity inequalities in the concentration-compactness method as introduced by the author [23], [24]. See also for some extensions, or applications of these arguments: M.J. Esteban and P.L. Lions [19], [20], M.J. Esteban [17], [18], M. Weinstein [31], P.L. Lions [25], H. Berestycki and P.L. Lions [9], A. Bahri and P.L. Lions [4], P.L. Lions [26], D. Gogny and P.L. Lions [21]. Roughly speaking, this method shows that, for various minimization problems in unbounded domains with constraints, all minimizing sequences are converging if and only if a certain strict subadditivity inequality holds. This inequality involves the infimum of the minimization problem as a function of the “level of the constraint” and corresponds to the possible losses of compactness which are basically due to the effect of “unbounded translations” or “concentrating-diluting dilations.” In some sense, the strict subadditivity inequalities represent the energy balance preventing (and this is necessary and sufficient) losses of compactness. The subadditivity comes into the picture because losses of compactness, when they occur, split the minimizing sequences into various parts which are either “infinitely away from each other” (translations) or “live in different scales” (dilations).

Consultant at CEA — Service de Physique Générale