Viscosity solutions of nonlinear second order elliptic PDEs involving nonlocal operators
Nonlinear partial differential equations of the form \(F(x,u, Du, D^ 2u, u - Mu) = 0\) in \(\Omega\) and \(\beta (x,u,u - Mu) = 0\) on \(\partial \Omega\), where \(M\) is a nonlocal operator, are studied in terms of viscosity solutions. Using monotonicity, ellipticity and local Lipschitz conditions on \(F\) it is shown that if there is an u.s.c. (viscosity) subsolution and a l.s.c. (viscosity) supersolution, then a unique solution exists. The results are applied to two model problems \[ \max \{Lu - f,u - \varphi \} = 0 \quad \text{in} \quad \Omega, \quad u(x) = \int_ \Omega u(y) Q(dy,x),\;x \in \partial \Omega, \] where \(L\) is an integro-differential operator and \[ \max \{Lu - f,u - Mu\} = 0 \quad \text{in} \quad \Omega, \quad \max \{u - g,u - Mu\} = 0 \quad \text{on} \quad \partial \Omega, \] where \(L\) is a second order elliptic operator.
scientific article; zbMATH DE number 579206