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Of all publications in the section: 5
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Article
Mariani M., Yannick S. Calculus of Variations and Partial Differential Equations. 2013. Vol. 46. No. 3-4. P. 687-703.

We are concerned with a control problem related to the vanishing fractional viscosity approximation to scalar conservation laws. We investigate the Γ-convergence of the control cost functional, as the viscosity coefficient tends to zero.

Article
Bogachev V., Shaposhnikov A., Shaposhnikov S. Calculus of Variations and Partial Differential Equations. 2019. Vol. 58. No. 5(176). P. 1-16.

Log-Sobolev-type inequalities for solutions to stationary Fokker-Planck-Kolmogorov equations are obtained.

Article
Kolesnikov A., Gladkov N., Zimin A. Calculus of Variations and Partial Differential Equations. 2019. Vol. 58. No. 173. P. 1-33.

The multistochastic (nk)-Monge–Kantorovich problem on a product space ∏ni=1Xi∏i=1nXi is an extension of the classical Monge–Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×⋯×XikXi1×⋯×Xik for all k-tuples {i1,…,ik}⊂{1,…,n}{i1,…,ik}⊂{1,…,n} for a given 1≤k<n1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution ππ to the following important model case: n=3,k=2,Xi=[0,1]n=3,k=2,Xi=[0,1], the cost function c(x,y,z)=xyzc(x,y,z)=xyz, and the corresponding two-dimensional projections are Lebesgue measures on [0,1]2[0,1]2. We prove, in particular, that the mapping (x,y)→x⊕y(x,y)→x⊕y, where ⊕⊕ is the bitwise addition (xor- or Nim-addition) on [0,1]≅Z∞2[0,1]≅Z2∞, is the corresponding optimal transportation. In particular, the support of ππ is the Sierpiński tetrahedron. In addition, we describe a solution to the corresponding dual problem.

Article
Kolesnikov A., Tikhonov S. Y. Calculus of Variations and Partial Differential Equations. 2014. Vol. 49. No. 3-4. P. 1187-1197.

Let \mu = e^{-V} \ dx be a probability measure and T = \nabla \Phi be the optimal transportation mapping pushing forward \mu onto a log-concave compactly supported measure \nu = e^{-W} \ dx. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi)} in the Besov spaces W^{\gamma,1}_{loc}. We prove that D^2 \Phi \in W^{\gamma,1}_{loc} provided e^{-V} belongs to a proper Besov class and W is convex. In particular, D^2 \Phi \in L^p_{loc} for some p>1. Our proof does not rely on the previously known regularity results.

Article
Kolesnikov A., Milman E. Calculus of Variations and Partial Differential Equations. 2016. Vol. 55. P. 1-36.

Given a probability measure μ supported on a convex subset of Euclidean

space (Rd , g0), we are interested in obtaining Poincaré and log-Sobolev type inequalities

on (, g0,μ). To this end, we change the metric g0 to a more general Riemannian one g,

adapted in a certain sense toμ, and perform our analysis on (, g,μ). The types ofmetrics we

consider are Hessian metrics (intimately related to associated optimal-transport problems),

productmetrics (which are very usefulwhenμis unconditional, i.e. invariant under reflections

with respect to the coordinate hyperplanes), and metrics conformal to the Euclidean one,

which have not been previously explored in this context. Invoking on (, g,μ) tools such as

Riemannian generalizations of theBrascamp–Lieb inequality and the Bakry–Émery criterion,

and passing back to the original Euclidean metric, we obtain various weighted inequalities on

(, g0,μ): refined and entropic versions of theBrascamp–Lieb inequality,weighted Poincaré

and log-Sobolev inequalities, Hardy-type inequalities, etc.Key to our analysis is the positivity

of the associated Lichnerowicz–Bakry–Émery generalized Ricci curvature tensor, and the

convexity of themanifold (, g,μ). In some cases,we can only ensure that the lattermanifold

is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.