The Morse-Sard theorem in Sobolev spaces,

Let Ω be an open subset of R n . Consider a differentiable map u : Ω → R m . For many application in differential topology, dynamical systems, and degree theory, it is important to study the “size” of the set of critical values of u . Usually the word “size” we just used is intended in the sense of some measure (e.g. Hausdorff measure, Lebesgue measure, entropy measure). The Morse-Sard Theorem is concerned exactly about the size of such set. To be precise, we will indicate by C u the set of the critical points of u (i.e., the set of points x ∈ Ω such that ∇ u ( x ) is not of maximum rank), and by V u the set u ( C u ) which is by definition the set of the critical values of u . In this paper we will prove that, if u ∈ W loc k , p ( Ω , R m ) for k = n − m + 1 , n < p , then the set of the critical value of u has m -measure zero. As we are dealing with a very classical theorem, we find it suitable to give an account with discussed bibliography of what is already known about the finite dimensional Morse-Sard theorem. Along the paper we will make the suitable comparisons.