Continuous dependence estimates and homogenization of quasi-monotone systems of fully nonlinear second order parabolic equations
The authors consider weakly coupled systems of parabolic equations written as \(\partial _{t}u_{i}+H_{i}(t,x,u,Du_{i},D^{2}u^{i})=0\) in \((0,T)\times \mathbb{R}^{m}\) for \(i=1,\cdots ,m\). Here \(H_{i}\) is an Hamiltonian operator and \(u(x)=(u_{1}(x),\cdots ,u_{m}(x))\). Assumptions are imposed on the data of the problem, especially on the Hamiltonian operator which lead to a quasi-monotonicity property of the system. The authors then define the notion of viscosity solution for such systems. The main result of the paper proves a continuous dependence estimate on the solution of such quasi-monotone systems, that is an estimate on the difference between the solutions of two systems with different Hamiltonian operators, in terms of the corresponding data. The authors finally apply the tools they use in the proof of the continuous dependence estimate in different situations. They start with estimates on the solution of a quasi-monotone system under some further hypotheses on the data. They go on proving estimates on the vanishing viscosity approximation of a first-order problem. In the last part of the paper, they describe a periodic homogenization result for quasi-monotone and fully nonlinear systems.
scientific article; zbMATH DE number 6057293
scientific article