Abstract: The talks are based on the results of our joint papers, some of new results were obtained recently in collaboration with Adele Ferone and Alba Roviello from the University of Caserta, Italy. We prove a bridge theorem that includes the classical Morse–Sard Theorem on critical values and its previous elegant extensions obtained by Dubovitskii and Federer as particular cases: namely, if a function v : Rn → Rd belongs to the Holder class C^{k,α}, 0 ≤ α ≤ 1, then for every q > m − 1 the identity H^µ(Zv,m ∩ v^{−1}(y)) = 0 holds for H^q-almost all y ∈ Rd, where Zv,m = {x ∈ Rn: rank ∇v(x) < m} is a m-critical set and µ = n − m + 1 − (k + α)(q − m + 1). Intuitively, the sense of this bridge theorem is very close to Heisenberg’s uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical Ck-case (when α = 0 ). These results helped for the solution of the so-called Leray’s problem in mathematical fluid mechanics. The problem remained open for more than 80 years (starting from the publication of the famous paper by Jean Leray 1933 [5] ). Namely, for plane and axially symmetric spatial flows the existence theorem was proved [3] for boundary value problem of stationary Navier-Stokes equations in bounded domains under necessary and sufficient condition of zero total flux (see also [4] for the case of exterior (unbounded) domains).

Cite: Korobkov M.V.
A bridge between Morse-Sard-Dubovitskii-Federer theorems and applications in fluid mechanics
The 41st Summer Symposium in Real Analysis (The Wooster -- The Black Squirrel Symposium) 18-24 Jun 2017