On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients

UDC 519.21 In this paper we solve a selection problem for multidimensional SDE where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane It is assumed that the drift has a Hoelder asymptotics as approaches and the limit ODE does not have a unique solution.We show that if the drift pushes the solution away from then the limit process with certain probabilities selects some extremal solutions to the limit ODE. If the drift attracts the solution to then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.

Regularity for scalar integrals without structure conditions

Integrals of the Calculus of Variations with {p,q}-growth may have not smooth minimizers, not even bounded, for general {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand {f(x,\xi)} with dependence on the modulus of the gradient, i.e. {f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if {\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous.

New Constraint Qualifications for S-Stationarity for MPEC with Nonsmooth Objective

This paper considers a mathematical problem with equilibrium constraints (MPEC) in which the objective is locally Lipschitz continuous but not continuously differentiable everywhere. Our focus is on constraint qualifications for the nonsmooth S-stationarity in the sense of the limiting subdifferentials. First, although the MPEC-LICQ is not a constraint qualification for the nonsmooth S-stationarity, we show that the MPEC-LICQ can serve as a constraint qualification for the nonsmooth S-stationarity under some kind of regularity. Then, we extend some new constraint qualifications for nonlinear programs to the considered nonsmooth MPEC and show that all of them can serve as constraint qualifications for the nonsmooth S-stationarity. We further extend these results to the multiobjective case.

Multiplicity of solutions to discrete inclusions with the p(k)-Laplace type equations

In this article, we prove the existence and multiplicity of solutions to discrete inclusions with the p(k)-Laplace type equations. We are interested in the existence of three solutions with the aid of linking arguments and using a three critical points theorem, for locally Lipschitz continuous fonctions.