The theory of first-order mean field type differential games examines the systems infinitely many identical agents interacting via some external media under assumption that each agent is controlled by two players. We study approximations value function game using solutions model finite-dimensional games. appears as a continuous-time Markov game, i.e., theoretical problem with and dynamics deter...

We consider a model for the motion of phase interface in an elastic medium, example, twin boundary martensite. The is given by semilinear parabolic equation with fractional Laplacian as regularizing operator, stemming from interaction front its environment. We show that presence randomly distributed, localized obstacles leads to threshold phenomenon, i.e., stationar...

In general, the value function associated with an exit time problem is a discontin-uous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the Hamilton-Jacobi equation involving the proximal subdiierentials (superdiierentials) with subdiierential type (superdiierential type) mixed boundary condition. We also show that if t...

We consider a robust switching control problem. The controller only observes the evolution of the state process, and thus uses feedback (closed-loop) switching strategies, a nonstandard class of switching controls introduced in this paper. The adverse player (nature) chooses open-loop controls that represent the so-called Knightian uncertainty, i.e., misspecifications of the model. The (half) g...

A comparison principle for the subdiffusive p-Laplacian in a possibly nonsmooth and unbounded open set is proved. The result requires that the involved sub and supersolution are positive, and the ratio of the former to the latter is bounded. As an application, constrained radial symmetry for overdetermined problems is obtained. More precisely, both Dirichlet and Neumann conditions are prescribe...

We study general 2nd order fully nonlinear degenerate elliptic equations on an arbitrary closed set with generalized Dirichlet boundary conditions in the viscosity sense. We prove some properties of the maximal subsolution and the minimal supersolution of the Dirichlet type problem. Under a sort of compatibility condition on the boundary data we show that the maximal subsolution is the natural ...