Weak Solutions for Obstacle Problems with Weak Monotonicity

This paper is concerned with the existence of weak solutions for obstacle problems. By means of the Young measure theory and a theorem of Kinderlehrer and Stampacchia, we obtain the needed result.

Directional Shift-Stable Functions

Recently, some new types of monotonicity—in particular, weak monotonicity and directional monotonicity of an n-ary real function—were introduced and successfully applied. Inspired by these generalizations of monotonicity, we introduce a new notion for n-ary functions acting on [0,1]n, namely, the directional shift stability. This new property extends the standard shift invariantness (difference scale invariantness), which can be seen as a particular directional shift stability. The newly proposed property can also be seen as a particular kind of local linearity. Several examples and a complete characterization for the case of n=2 of directionally shift-stable aggregation and pre-aggregation functions are also given.

On strongly quasilinear elliptic systems with weak monotonicity

In this paper, we prove existence results in the setting of Sobolev spaces for a strongly quasilinear elliptic system by means of Young measures and mild monotonicity assumptions.

Mean-field backward–forward stochastic differential equations and nonzero sum stochastic differential games

We study a general class of fully coupled backward–forward stochastic differential equations of mean-field type (MF-BFSDE). We derive existence and uniqueness results for such a system under weak monotonicity assumptions and without the non-degeneracy condition on the forward equation. This is achieved by suggesting an implicit approximation scheme that is shown to converge to the solution of the system of MF-BFSDE. We apply these results to derive an explicit form of open-loop Nash equilibrium strategies for nonzero sum mean-field linear-quadratic stochastic differential games with random coefficients. These strategies are valid for any time horizon of the game.

Backward Doubly SDEs with weak Monotonicity and General Growth Generators

We deal with backward doubly stochastic differential equations (BDSDEs) with a weak monotonicity and general growth generators and a square integrable terminal datum. We show the existence and uniqueness of solutions. As application, we establish the existenceand uniqueness of Sobolev solutions to some semilinear stochastic partial differential equations (SPDEs) with a general growth and a weak monotonicity generators. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.