A priori estimates and existence of solutions to the prescribed centroaffine curvature problem

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space ${\mathbb{R}}^{n+1}$. This problem is equivalent to solving a Monge–Ampère equation on the unit sphere. It corresponds to the critical case of the Blaschke–Santaló inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.     

Existence for translating solutions of Gauss curvature flow on exterior domains

In this paper, we use the Perron method to prove the existence of viscosity solutions to a class of Monge–Ampère equations on exterior domains in ${\mathbb{R}}^n(n\ge 2)$ with prescribed asymptotic behavior at infinity. This problem comes from the study of Gauss curvature flow and its generalization, the flow by powers of Gauss curvature.     

Mayer control problem with probabilistic uncertainty on initial positions

In this paper we introduce and study an optimal control problem in the Mayer's form in the space of probability measures on ${\mathbb{R}}^n$ endowed with the Wasserstein distance. Our aim is to study optimality conditions when the knowledge of the initial state and velocity is subject to some uncertainty, which are modeled by a probability measure on ${\mathbb{R}}^d$ and by a vector-valued measure on ${\mathbb{R}}^d$, respectively. We provide a characterization of the value function of such a problem as unique solution of an Hamilton–Jacobi–Bellman equation in the space of measures in a suitable viscosity sense. Some applications to a pursuit-evasion game with uncertainty in the state space is also discussed, proving the existence of a value for the game.     

G-Ham Sandwich Theorems: Balancing measures by finite subgroups of spheres

Equivariant Ham Sandwich Theorems are obtained for the classical algebras $\mathbb{F}=\mathbb{R},\mathbb{C}\text{, and}\mathbb{H}$ and the finite subgroups G of their unit spheres. Given any n $\mathbb{F}$-valued Borel measures on ${\mathbb{F}}^n$ and any n-dimensional free $\mathbb{F}$-unitary representation of G, it is shown that there exists a Voronoi partition of ${\mathbb{F}}^n$ naturally determined by G which “G-balances” each measure, as realized by the simultaneous vanishing of each “G-average” of the measures of the partition?s isometric fundamental domains. Applications for real measures follow, among them that any n signed mass distributions on ${\mathbb{C}}^{(p?1)n/2}$ can be equipartitioned by a single complex regular p-fan if p is an odd prime.     

LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres

Let ${\mathbb{S}}^d$ denote the unit sphere in the Euclidean space ${\mathbb{R}}^{d+1}(d\ge 1)$. We develop LeVeque type inequalities for the discrepancy between the rotationally invariant probability measure and the normalized counting measures on ${\mathbb{S}}^d$. We obtain both upper bound and lower bound estimates. We then use these inequalities to estimate the discrepancy of the normalized counting measures associated with minimal energy configurations on ${\mathbb{S}}^d$.     

Sharp bounds for Hardy operators on product spaces

In this article, we obtain the sharp bounds from L^P$\left({\mathbb{G}}^n\right)$ to the space wL^P$\left({\mathbb{G}}^n\right)$ for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from L^P$\left({\mathbb{G}}^n\right)$ to the space L^P_I$\left({\mathbb{G}}^n\right)$ are obtained.     

Enumerating invariant subspaces of Rn

In this article, we develop an algorithm to calculate the set of all integers m for which there exists a linear operator T on ${\mathbb{R}}^n$ such that ${\mathbb{R}}^n$ has exactly m T-invariant subspaces. Moreover, the algorithm gives a method for the explicit construction of such a linear operator T. A brief discussion is included as how these methods can be extended to vector spaces over arbitrary fields.