Jens Frehse

Mean field games and mean field type control theory

Introduction.- General Presentation of Mean Field Control Problems.- Discussion of the Mean Field game.- Discussion of the Mean Field Type Control.- Approximation of Nash Games with a large number of players.- Linear Quadratic Models.- Stationary Problems- Different Populations.- Nash differential games with Mean Field effect.

Regularity results for nonlinear elliptic systems and applications

General Technical Results.- General Regularity Results.- Nonlinear Elliptic Systems Arising from Stochastic Games.- Nonlinear Elliptic Systems Arising from Ergodic Control.- Harmonic Mappings.- Nonlinear Elliptic Systems Arising from the Theory of Semiconductors.- Stationary Navier-Stokes Equations.- Strongly Coupled Elliptic Systems.- Dual Approach to Nonlinear Elliptic Systems.- Nonlinear Elliptic Systems Arising from plasticity Theory.

On Analysis of Steady Flows of Fluids with Shear-Dependent Viscosity Based on the Lipschitz Truncation Method

We deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for

On the Regularity of the Solution of the Biharmonic Variational Inequality.

p>\frac{2d}{d+2}

ERGODIC BELLMAN SYSTEMS FOR STOCHASTIC GAMES IN ARBITRARY DIMENSION

. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.

On two-dimensional quasi-linear elliptic systems

It is shown that the solution of the biharmonic variational inequality has bounded second derivatives provided that the obstacle and the data are smooth.

Large Data Existence Result for Unsteady Flows of Inhomogeneous Shear-Thickening Heat-Conducting Incompressible Fluids

Stochastic games with cost functionals J(i) ρ, x(v) = E ∫∞ 0 e–ρtli(y, v) dt, i = 1, 2 with controls v = v1, v2 and state y(t) with y(0) = x are considered. Each player wants to minimize his (her) cost functional. E denotes the expected value and the state variables y are coupled with the controls v via a stochastic differential equation with initial value x. The corresponding Bellman system, which is used for the calculation of feedback controls v = v(y) and the solvability of the game, leads to a class of diagonal second-order nonlinear elliptic systems, which also occur in other branches of analysis. Their behaviour concerning existence and regularity of solutions is, despite many positive results, not yet well understood, even in the case where the li, are simple quadratic functions. The objective of this paper is to give new insight to these questions for fixed ρ > 0, and, primarily, to analyse the limiting behaviour as the discount ρ → 0. We find that the modified solutions of the stochastic games converge, for subsequences, to the solution of the so-called ergodic Bellman equation and that the average cost converges. A former restriction of the space dimension has been removed. A reasonable class of quadratic integrands may be treated. More specifically, we consider the Bellman systems of equations – ∆z + λ = H (x, Dz), where the space variable x belongs to a periodic cube (for the sake of simplifying the presentation). They are shown to have smooth solutions. If uρ is the solution of – ∆uρ + ρuρ = H (x, Du ρ) then the convergence of uρ — ῡρ to z, as ρ tends to 0, is established. The conditions on H are such that some quadratic growth in Du is allowed.

Regularity for the stationary Navier-Stokes equations in bounded domain

AbstractThe author shows the existence of a Hölder continuous solution for a class of two-dimensional non-linear elliptic systems of the type

On a Stokes-Like System for Mixtures of Fluids

- \Sigma _{i = 1}^2 \partial _i a_i (x,u,\triangledown u) + a_o (x,u,\triangledown u) = 0.