Max-min representation of solutions of Hamilton-Jacobi equation in the quarter plane
The monotone difference scheme \(u_k^n:=u(k\Delta x,n\Delta t)\), \(k=0,\pm 1,\pm 2,\dots\), \(n=0,1,\dots\), \(\Delta x=L\Delta t\rightarrow 0\), is used to obtain max-min representations of generalized solutions of Hamilton-Jacobi equations of the form: \(u_t+H(u_x)=0\) defined by Hamiltonians \(H:\mathbb{R}^n\rightarrow\mathbb{R}\) that are assumed to be \(L\)-Lipschitzian. Iterating the recurrence formula: \[ u_k^n=\max_{\alpha\in\mathbb{R}}\min_{\beta=\pm 1}[u_{k-\beta}^{n-1}+\Delta t(L\alpha\beta-H(\alpha))], \] the authors obtain \(u_k^n\) as a max-min function of the data \(u_k^0:=u^0(k\Delta x)\), \(u_0^n:=\lambda(n\Delta t)\) and therefore a max-min representation of the limit function: \(u(x,t):=\lim_{\Delta t\rightarrow 0}u_k^n\) if \(x=k\Delta x=kL\Delta t\), \(t=n\Delta t\), which is considered a generalized solution of the initial value problem: \(u(x,0)=u^0(x)\), \(x\in\mathbb{R}\), \(t>0\), of the Dirichlet problem: \(u(x,0)=u^0(x)\), \(u(0,t)=\lambda(t)\), \(x,t>0\) and, respectively, of the Neumann problem: \(u(x,0)=u^0(x)\), \(u_x(0,t)=\lambda(t)\), \(x,t>0\).
max-min representations of generalized solutions
scientific article; zbMATH DE number 1014374