Existence of Optimal Solutions for Learning-based PDEs Governed by Time-Dependent p-Laplace and 1-Laplace Operators

Abstract

Learning-based partial differential equations, with applications in image processing and computer vision, are often formulated as optimal control problems constrained by PDEs. In this paper, we consider PDE constraints governed by the combination of the time-dependent p-Laplace and singular 1-Laplace type operators. Models of this type–and more general formulations–have been proposed and investigated numerically in Liu et al. [15, 16]. Theoretical results on the existence of optimal solutions have so far been established only for constant-coefficient operators in recent work [17]. The extension to time-dependent coefficients is more delicate, due to the lack of suitable stability results for nonlinear semigroups generated by the sum of two time-dependent monotone operators, together with the degeneracy of the multi-valued 1-Laplace operator. The main contribution of this paper is to address these issues and to establish the existence of optimal solutions for a class of singular PDE-constrained optimal control problems governed by the sum of time-dependent p-Laplace and multi-valued 1-Laplace operators.