Total $$\mathbb {A}$$-variation flows
Abstract
Abstract We study the $$L^2$$ -gradient flows, $$\partial _t u-\operatorname {div}(\textrm{D}f(x,\mathbb {A}u))=0$$ , of functionals of the type $$\int _{\Omega }f(x,\mathbb {A}u)\,\textrm{d}x$$ , where f is a convex function of linear growth and $$\mathbb {A}$$ is some first-order linear constant-coefficient differential operator. To this end, we identify the relaxation of the functional to the space $$BV^\mathbb {A}\cap L^2$$ , identify its subdifferential, and show pointwise representation formulas for the relaxation and the subdifferential, both with and without Dirichlet boundary conditions. The existence and uniqueness then follow from abstract semigroup theory. We further show that our solutions can be obtained as limits of the corresponding flows with p -growth as $$p\searrow 1$$ .
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Geometric Analysis and Curvature Flows
Advanced Mathematical Modeling in Engineering
Publication details
- Journal / source
- Annali di Matematica Pura ed Applicata (1923 -)
- Publisher
- Springer Science and Business Media LLC
- Volume
- 205
- Issue
- 5
- Pages
- 2819-2856
- Publication date
- 6 May 2026
- Language
- EN