Effective Hamiltonian and homogenization of measurable eikonal equations
The authors describe the homogenization of evolution Hamilton-Jacobi equations \(u_{\varepsilon t}+H(Du_{\varepsilon })=f(x/\varepsilon )\) posed in \(\mathbb R^{N}\times (0,+\infty)\). Here the potential \(f\) is supposed to be \(\mathbb Z^{N}\)-periodic, measurable and essentially bounded. The function \(H\) satisfies the usual conditions of convexity and coercivity. The solution \( u_{\varepsilon }\) starts from a uniformly continuous data \(u_{0}\) at \(t=0\). The main result of the paper proves that \(u_{\varepsilon }\) converges locally and uniformly in \(\mathbb R^{N}\times (0,+\infty )\) to the solution of \( u_{t}+\overline{H}(Du)=0\) posed in \(\mathbb R^{N}\times (0,+\infty )\), where \( \overline{H}\) is the unique value of the parameter \(a\) which makes the cell problem \(H(Du+p_{0})=f(x)+a\) solvable by a periodic function, for every \( p_{0}\). The authors first define an appropriate notion of viscosity subsolution, supersolution and solution for the original problem. They prove some properties of these solutions and prove the existence of a unique uniformly continuous solution \(u_{\varepsilon }\) of the original problem. They then build the notion of viscosity solution for the cell problem, which leads to the existence of a unique viscosity solution of this cell problem. The last part of the paper proves the homogenization result. This is done using appropriate subsolutions and supersolutions.
scientific article; zbMATH DE number 5533119