Special Session 146: 

Equivalence of two definitions on generalized solutions of Dirichlet problems associated to fractional Laplacian operator

Qingshuo Song
City University of Hong Kong
Hong Kong
Co-Author(s):    Qingshuo Song
Probabilistic definition of the generalized solutions depends on fine topology of the associated stochastic process. In the context of PDE analysis, the generalized solution has been defined purely by viscosity superjet and subjet of differential operators, see for instance User`s guide in the viscosity solution in 1992. We will show these two definitions are equivalent under very mild conditions. Together with Perron`s method, the result can be used to prove solvability of HJB equations with Dirichlet boundary.