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Viscosity solutions to the inhomogeneous reaction–diffusion equation involving the infinity Laplacian
Preview AbstractIn this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian:
Δ∞u(x)=f(x,u(x))+g(x),x∈Ω,where the continuous function f satisfies 0≤f(x,δt)≤Λ(x)δγf(x,t), a positive function Λ(x)∈C(¯Ω),γ∈[0,3),t>0 and 0<δ≤12. Such a model permits existence of solutions with dead core zones, i.e. a priori unknown regions where non-negative solutions vanish identically. For γ∈[0,3) and the non-positive inhomogeneous term g, we establish the existence, uniqueness and stability of the viscosity solution of the corresponding continuous Dirichlet problem. Under additional structure conditions on f and g, we obtain the optimal C43−γ regularity across the free boundary ∂{u>0}∩Ω. Moreover, we establish the porosity of the free boundary and Liouville type theorem for entire solutions. Finally, we prove that the dead core vanishes in the limit case γ=3.On the interplay between nonlinear partial differential equations and game theory
Preview AbstractWe review some recent results concerning game theory and their relation to some well known PDEs. In particular, we will show that solutions to certain PDEs can be obtained as limits of values of tug-of-war games when the parameter that controls the length of the possible movements goes to zero.