87 research outputs found
Clarke subgradients of stratifiable functions
We establish the following result: if the graph of a (nonsmooth)
real-extended-valued function
is closed and admits a Whitney stratification, then the norm of the gradient of
at relative to the stratum containing bounds from below
all norms of Clarke subgradients of at . As a consequence, we obtain
some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz
inequality for functions definable in an arbitrary o-minimal structure
On the first integral conjecture of Rene Thom
More that half a century ago R. Thom asserted in an unpublished manuscript
that, generically, vector fields on compact connected smooth manifolds without
boundary can admit only trivial continuous first integrals. Though somehow
unprecise for what concerns the interpretation of the word \textquotedblleft
generically\textquotedblright, this statement is ostensibly true and is
nowadays commonly accepted. On the other hand, the (few) known formal proofs of
Thom's conjecture are all relying to the classical Sard theorem and are thus
requiring the technical assumption that first integrals should be of class
with where is the dimension of the manifold. In this
work, using a recent nonsmooth extension of Sard theorem we establish the
validity of Thom's conjecture for locally Lipschitz first integrals,
interpreting genericity in the sense
A family of functional inequalities
For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Lojasiewicz inequalities. In a second part, we specialise these inequalities to some classical geodesically convex functionals. For the Boltzmann entropy, we obtain the equivalence between logarithmic Sobolev and Talagrand's inequalities. On the other hand, the non-linear entropy and the Gagliardo-Nirenberg inequality provide a Talagrand inequality which seems to be a new equivalence. Our method allows also to recover some results on the asymptotic behaviour of the associated gradient flows
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