On a Neumann problem for variational functionals of linear growth

TL;DR

This paper proves the existence of minimizers for a class of convex variational problems with linear growth under Neumann boundary conditions, without needing relaxation to BV spaces, unlike the Dirichlet case.

Contribution

It establishes existence results for Neumann problems of convex functionals of linear growth directly in W^{1,1}, avoiding the need for BV relaxation, which was previously only known in more restrictive Dirichlet cases.

Findings

Existence of minimizers in W^{1,1} for Neumann problems.

No relaxation to BV needed for simply connected domains.

Comparison with Dirichlet problems highlights less restrictive conditions.

Abstract

We consider a Neumann problem for strictly convex variational functionals of linear growth. We establish the existence of minimisers among $\operatorname{W}^{1,1}$-functions provided that the domain under consideration is simply connected. Hence, in this situation, the relaxation of the functional to the space of functions of bounded variation, which has better compactness properties, is not necessary. Similar $\operatorname{W}^{1,1}$-regularity results for the corresponding Dirichlet problem are only known under rather restrictive convexity assumptions limiting its non-uniformity up to the borderline case of the minimal surface functional, whereas for the Neumann problem no such quantified version of strong convexity is required.