PreprintThe DirichlettoNeumann operator associated with the 1Laplacian and evolution problemsDaniel Hauer and José M. MazónAbstractIn this paper, we present the first insights about the DirichlettoNeumann operator in \(L^{1}\) associated with the \(1\)Laplace operator or total variational flow operator. This operator is the main object, for example, in studying inverse problems related to image processing, but also admits important relation to geometry. We show that this operator can be represented by the subdifferential in \(L^1\times L^{\infty}\) of a convex, homogeneous, and continuous functional on \(L^{1}\). This is quite surprising since it implies a type of stability or compactness result even though the singular Dirichlet problem governed \(1\)Laplace operator by the might have infinitely many weak solutions if the given boundary data is not continuous. As an application, we obtain wellposedness and longtime stability of solutions of a singular coupled ellipticparabolic initial boundaryvalue problem. Keywords: Subdifferential, nonlinear semigroups, \(L^1\) regularity theory, total variational flow, \(1\)Laplacian, least gradient, DirichlettoNeumann operator.AMS Subject Classification: Primary 35K65; secondary 35J25, 35J92, 35B40.
This paper is available as a pdf (404kB) file. It is also on the arXiv: arxiv.org/abs/arXiv:1910.12219.
