Viscosity solutions of monotone systems for Dirichlet problems
Existence of viscosity solutions for fully nonlinear elliptic systems of second order of the form \(F_ k (x, u, D_ k u, D^ 2 u_ k) =0\) for \(x\in G\), \(k= 1,\dots, m\), where \(G\) is a smooth bounded domain in \(\mathbb{R}^ N\) is proved in the case, where the system is monotone in a suitable sense. This extends some previous work by different people, including the authors. Definitions of multivalued sub- and supersolutions are first given and then a uniqueness theorem is proved. A comparison result for monotone systems is also proved and this allows to compare nontangential semicontinuous sub- and supersolutions. Finally the existence result is proved.
scientific article; zbMATH DE number 527354