Morse theory for some lower-C 2 functions in finite dimension

Abstract

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C2 functions. For this purpose, we first recall some properties of the class of lower-C2 functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a “Morse” index for a lower-C2 functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C2 functions.