Adjoint methods for obstacle problems and weakly coupled systems of PDE (Q2842252)

The authors give different applications of the adjoint method, which was introduced in [\textit{L. C. Evans}, Arch. Ration. Mech. Anal. 197, No. 3, 1053--1088 (2010; Zbl 1273.70030)]. In the following, \(U\subset {\mathbb{R}}^n\) is an open bounded domain with smooth boundary and \(n\geq 2\). They first study the obstacle problem NEWLINE\[NEWLINE\begin{cases}\max\{u-\psi,u+H(x,Du)\}=0,\text{ in }U, \\ u=0,\text{ on }\partial U,\end{cases}NEWLINE\]NEWLINE where \(\psi: \overline{U}\longrightarrow {\mathbb{R}}\) and \(H: {\mathbb{R}}^n\times \overline{U}\longrightarrow {\mathbb{R}}\) are smooth functions with \(\psi\geq 0\) on \(\partial U\). The authors then study the weakly coupled system of Hamilton-Jacobi equations NEWLINE\[NEWLINE\begin{cases} c_{11}u_1+c_{12}u_2+H_1(x,Du_1)=0,\text{ in }U,\\ c_{21}u_1+c_{22}u_2+H_2(x,Du_2)=0,\text{ in }U, \\u_1=u_2=0,\text{ on }\partial U.\end{cases}NEWLINE\]NEWLINE The third application concerns the cell problem NEWLINE\[NEWLINE\begin{cases} c_1u_1-c_1u_2+H_1(x,Du_1)=\overline{H}_1,\text{ in }{\mathbb{T}}^n,\\ -c_2u_1+c_2u_2+H_2(x,Du_2)=\overline{H}_2,\text{ in }{\mathbb{T}}^n,\end{cases}NEWLINE\]NEWLINE where \(u_1\) ,\(u_2: {\mathbb{T}}^n\longrightarrow {\mathbb{R}}\) and \(\overline{H}_1,\overline{H}_2\in {\mathbb{R}}\) are the unknowns. Finally, they study the weakly coupled systems of obstacle type NEWLINE\[NEWLINE\begin{cases}\max\{u_1-u_2-\psi_1,u_1+H_1(x,Du_1)\}=0,\text{ in }U,\\ \max\{u_2-u_1-\psi_2,u_2+H_2(x,Du_2)\}=0,\text{ in }U, \\ u_1=u_2=0,\text{ on }\partial U,\end{cases}NEWLINE\]NEWLINE where \(\psi_1,\psi_2: \overline{U}\longrightarrow {\mathbb{R}}\) and \(H_1,H_2: {\mathbb{R}}^n\times \overline{U}\longrightarrow {\mathbb{R}}\) are smooth functions with \(\psi_1,\psi_2\geq \alpha>0\). In each case, the authors obtain the convergence, with a precise rate, of the corresponding regularised procedure.