Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation
In the very interesting paper under review, the authors introduce a new approach to the Dirichlet problem for the total variation flow in a bounded domain and analyze the associated inhomogeneous problem. Precisely, the paper deals with the problem \[ \begin{cases} \dfrac{\partial u}{\partial t}- \operatorname{div} \left(\dfrac{Du}{|Du|}\right)=f(x,t) & \text{in } \Omega\times(0,+\infty),\\ u=0 & \text{on } \partial\Omega\times(0,+\infty),\\ u(x,0)=u_0(x) & \text{in } \Omega, \end{cases} \] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with Lipschitz continuous boundary, \(f\in L^1_{\text{loc}}(0,+\infty;L^2(\Omega))\), and \(u_0\in L^2(\Omega)\). The main result of the paper asserts the existence of a unique solution \(u\in L^1_{\text{loc}}(0,+\infty;BV(\Omega))\cap C([0,+\infty);L^2(\Omega))\) to the above problem. The authors compare, moreover, the solutions corresponding to different data and study the long-term behaviour of the solutions. Explicit examples of radial solutions are shown as well.
scientific article; zbMATH DE number 6494623