Trotter’s product formula for semigroups generated by quasilinear elliptic operators

Abstract:

Trotter’s product formula is given for nonlinear semigroups in ${L^1}({R^N})$ generated by quasilinear operators of the form $\Delta \phi$, where $\phi$ is a suitable function: formally $\exp (t\Delta \phi )u = {\lim _{h \downarrow 0}}{\left \{ {\exp (h\Delta {\phi _1}) \cdots \exp (h\Delta {\phi _k})} \right \}^{[t/h]}}u$, where $\phi = {\phi _1} + \cdots + {\phi _k}$. The proof is carried out by a new method for construction of a semigroup with generator $\Delta \phi$ in ${L^1}({R^N})$.