Acta Mathematica Vietnamica
Existence of Optimal Solutions for Learning-based PDEs Governed by Time-Dependent $p$-Laplace and 1-Laplace Operators
Van Thanh Nguyen
,
Thuan Quang Thai
,
Minh Hien Huynh
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 [
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