1,721,024 research outputs found
Il teorema di Morse-Sard in spazi di Sobolev. Problemi di trasporto ottimale ed applicazioni
No full textLimits of Inner Superposition Operators and Young Measures
No full textWe apply some well known theorems from the theory of Young's measures to the theory
of inner superposition (composition) operators.
We give an explicit characterization of the limit operator of a weak-convergent
sequence of inner superposition operators between Lebesgue spaces
The Morse-Sard theorem in Sobolev spaces,
No full textLet 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
The Monge problem for strictly convex norms in R^N
No full textWe prove the existence of an optimal transport map for the Monge problem in a convex bounded subset of Rd under the assumptions that the first marginal is absolutely continuous with respect to the Lebesgue measure and that the cost is given by a strictly convex norm. We propose a new approach which does not use disintegration of measures
Weak and strong density of compositions
No full textThe convergence in various topologies of sequences of inner superposition (composition) operators acting between Lebesgue spaces and of their linear combinations is studied. In particular, the sequential density results for the linear span of such operators is proved for the weak, weak continuous and strong operator topologies
Optimal transport with Coulomb cost. Approximation and duality
Full text linkWe revisit the duality theorem for multimarginal optimal transportation problems. In particular, we focus on the Coulomb cost. We use a discrete approximation to prove equality of the extremal values and some careful estimates of the approximating sequence to prove existence of maximizers for the dual problem (Kantorovich's potentials). Finally we observe that the same strategy can be applied to a more general class of costs and that a classical results on the topic cannot be applied here
Principles of comparison with distance functions for absolute minimizers
No full textWe extend the principle of comparison with cones introduced by Crandall, Evans and Gariepy in [12] for the Minimizing Lipschitz Extension Problem to a wide class of supremal functionals. This gives a geometrical characterization of the absolute minimizers (optimal solutions whose minimality is local). Some application to the stability of absolute minimizers with respect to the Gamma-convergence is given. A variation of the basic idea also allows to characterize the minimal Lipschitz extensions in length metric spaces
Asymptotic behavior of non linear eigenvalue problems involving Laplacian type operators
No full textWe study the asymptotic behaviour of two nonlinear eigenvalue problems which involve p-Laplacian-type operators. In the first problem we consider the limit as p goes to infinity of the sequences of the kth eigenvalues of the p-Laplacian operators. The second problem we study is the homogenization of nonlinear eigenvalue problems for some p-Laplacian-type operators with p fixed. Our asymptotic analysis relies on a convergence result for particular critical values of a class of Rayleigh quotients, stated in a unified framework, and on the notion of Gamma-convergence
Topological equivalence of some variational problems involving distances
No full textTo every distance d on a given open set \Omega\subseteq\mathbb R^n, we may associate several kinds of variational problems. We show that, on the class of all geodesic distances d on \Omega which are bounded from above and from below by fixed multiples of the Euclidean one, the uniform convergence on compact sets turns out to be equivalent to the \Gamma-convergence of each of the corresponding variational problems under consideration
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