Dual extrapolation and its applications to solving variational inequalities and related problems

In this paper we suggest new dual methods for solving variational inequalities with monotone operators. We show that with an appropriate step-size strategy, our method is optimal both for Lipschitz continuous operators ( $O({1 \over \epsilon})In this paper we suggest new dual methods for solving variational inequalities with monotone operators. We show that with an appropriate step-size strategy, our method is optimal both for Lipschitz continuous operators ( iterations), and for the operators with bounded variations ( iterations). Our technique can be applied for solving non-smooth convex minimization problems with known structure. In this case the worst-case complexity bound is iterations. The research results presented in this paper have been supported by a grant “Action de recherche concertè ARC 04/09-315” from the “Direction de la recherche scientifique, Communautè fran?aise de Belgique”. The scientific responsibility rests with its author(s).     

Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

We find necessary density conditions for Marcinkiewicz–Zygmund inequalities and interpolation for spaces of spherical harmonics in with respect to the L^p norm. Moreover, we prove that there are no complete interpolation families for p≠2.     

A remark on regularity criterion for the dissipative quasi-geostrophic equations

This paper concerns with a regularity criterion of solutions to the 2D dissipative quasi-geostrophic equations. Based on a logarithmic Sobolev inequality in Besov spaces, the absence of singularities of θ in [0,T] is derived for θ a solution on the interval [0,T) satisfying the condition

     

The surface measure and cone measure on the sphere of

We prove a concentration inequality for the norm on the sphere for . This inequality, which generalizes results of Schechtman and Zinn (2000), is used to study the distance between the cone measure and surface measure on the sphere of . In particular, we obtain a significant strengthening of the inequality derived by Naor and Romik (2003), and calculate the precise dependence of the constants that appeared there on .

     

Some integral mean estimates for polynomials

In this paper we establish Lq inequalities for polynomials, which in particular yields interesting generalizations of some Zygmund-type inequalities.     

Strict Feasibility of Variational Inequalities in Reflexive Banach Spaces

Strict feasibility is proved to be an equivalent characterization of （dual） variational inequalities having a nonempty bounded solution set, provided the mappings involved are stably properly quasimonotone. This generalizes an earlier result from finite-dimensional Euclidean spaces to infinitedimensional reflexive Banach spaces. Moreover, the monotonicity-type assumptions are also mildly relaxed.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

Mayer and optimal stopping stochastic control problems with discontinuous cost

We study two classes of stochastic control problems with semicontinuous cost: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient functions. Both the lower and upper semicontinuous cases are considered. The value function is shown to be a generalized viscosity solution of the associated HJB system, respectively, of some variational inequality. Dual formulations are given, as well as the relations between the primal and dual value functions. Under classical convexity assumptions, we prove the equivalence between the linearized Mayer problem and the standard weak control formulation. Counter-examples are given for the general framework.     

Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Existence, uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained. The nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions. The lower bound conditions are allowed to contain the critical Sobolev exponents. The methodology is to establish variational inequality principles for demicontinuous pseudo-contractive maps in Hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities. Examples on the existence, uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given.     

Optimal general inequalities for Lagrangian submanifolds in complex space forms

Lagrangian submanifolds appear naturally in the context of classical mechanics. Moreover, they play some important roles in supersymmetric field theories as well as in string theory. In this paper we establish general inequalities for Lagrangian submanifolds in complex space forms. We also provide examples showing that these inequalities are the best possible. Moreover, we provide simple non-minimal examples which satisfy the equality case of the improved inequalities.