A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations (Q850174)

The paper deals with viscosity solutions for a class of integro-differential equations in a bounded open domain in \(\mathbb{R}^n\). The author introduces new definitions for viscosity solutions (viscosity supersolution and viscosity subsolution), using the second order superjet (respectively, subjet) in the non-local term. By using these definitions, one avoids the singularity of the Lévy measure. Comparison results for the viscosity solutions for the Dirichlet and the Neumann boundary value problems are stated and proved here. Also, based on the Perron's method, the author proves the existence of the viscosity solutions for these boundary value problems.