Convex duality and uniqueness for BV-minimizers

Abstract

There are two different approaches to the Dirichlet minimization problem for variational integrals with linear growth. On the one hand, one commonly considers a generalized formulation in the space of functions of bounded variation. On the other hand, there is a closely related maximization problem in the space of divergence-free bounded vector fields, namely the dual problem in the sense of convex analysis.In this paper, we extend previous results on the duality correspondence between the generalized and the dual problem to a full characterization of their extremals via pointwise extremality relations. Furthermore, we discuss related uniqueness issues for both kinds of solutions and their relevance in the regularity theory of generalized minimizers.Our approach is sufficiently general to cover arbitrary dimensions, non-smooth integrands, and unbounded, irregular domains. MSC (2010): 49N15 (primary); 26B25, 26B30, 49K20 (secondary)

Functionals on Bv Space With Carath\'{e}odory Integrands Using Convex Duality

“…thus showing that Ω ϕ(x, Du) as defined in this paper and as defined from [2], [3], and [8] are equal under certain conditions for ϕ and ϕ * .…”

“…The definition we use here is in contrast to that which is used in works such as [2], [3], and [8], where continuity or lower semicontinuity of ϕ in both x and p are assumed for some of the same results obtained here, notably the existence of minimizers as proved in the next section. As in the anisotropic case above, lower semicontinuity of Ω ϕ(x, Du) in L 1 will easily follow in our case.…”

“…We further remark that convex duality is also used in Lemma 1 of [19] for Ω ϕ(x, Du) but with the requirement that ϕ ∈ C 1 (Ω × R N ) so as to use the implicit function theorem, and in [5] but in a more general functional analytic and measure theoretic setting. More recently, convex duality is also used in [8] to prove relations between generalized minimizers over BV (Ω) of Ω ϕ(x, Du) for linear growth convex ϕ in p and its dual problem with divergence free vector fields. The authors of [8] also, however, incorporate the theory as used in [2], [3].…”

“…More recently, convex duality is also used in [8] to prove relations between generalized minimizers over BV (Ω) of Ω ϕ(x, Du) for linear growth convex ϕ in p and its dual problem with divergence free vector fields. The authors of [8] also, however, incorporate the theory as used in [2], [3].…”

Strict interior approximation of sets of finite perimeter and functions of bounded variation

“…In fact, the extension of Theorem 1.1 (and of Theorem 1.2 stated below) to unbounded Ω is discussed in [5,Section 3.3].…”