An existence result for optimal control of singular constraints involving 1-Laplacian operator

Learning-based partial differential equations (PDEs), which arise from applications in image processing and computer vision, are optimal control problems governed by PDEs. In this paper, we show the maximal monotonicity of the sum operator of the p-Laplacian and the singular 1-Laplacian. The effective domain of the sum operator is shown to be dense in Hilbert space L-2(Omega) equipped with the norm topology. As an application, we derive the existence of optimal solutions to optimal control of singular PDEs involving the sum operator of the p-Laplacian and 1-Laplacian in the continuous setting.