On the Dirichlet problem for variational integrals in BV

Abstract

We investigate the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. We prove uniqueness of minimizers up to additive constants and deduce additional assertions about these constants and the possible (non-)attainment of the boundary values. Moreover, we provide several related examples. In the case of the model integral Ð W ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ j'wj 2 q dx for w : R n I W ! R N our results extend classical results from the scalar case N ¼ 1-where the problem coincides with the non-parametric least area problem-to the general vectorial setting N A N.

Lipschitz Bounds and Nonuniform Ellipticity

“…Here we need to remark that the arguments in [26, lemmas 3.1 and 3.2] work in a more general situation, where no upper bound on @ 2 F as in (8.2) is in force. Accordingly, these lemmas yield (8.5) without the upper bound on @ 2 F k displayed in (8.5) 3 . Essentially, what we can do here is use [26, lemmas 3.1 and 3.2] with the choice p D q (with the notation used in [26]; these correspond to in the present setting) and (8.2) 2;3 being in force.…”

“…Providing a general theory going beyond the validity of (1.39) is an interesting issue that will be treated in forthcoming work. We refer to [3,4] and related references for results in this direction.…”

Convex duality and uniqueness for BV-minimizers

“…Clearly, the latter assertion trivially implies uniqueness of the full gradient Du whenever one can prove W 1,1 -regularity for all generalized minimizers. While in [7] we have treated a borderline case of this regularity problem, we here discuss less subtle situations where it can be resolved -in a simpler and more elegant way -via Theorem 2.4. In particular, this happens in the following corollaries, which have previously been obtained -under stronger assumptions on f and based on a different strategy from [36,13] …”

“…On the one hand, we believe that this approach to uniqueness is more conceptual and less technical than the direct proof, implemented in [7], of W 1,1 -regularity for all minimizers. On the other hand, Corollary 2.6 remains limited to situations where C 1 -regularity is within reach, while the corollary does not seem to be applicable in the borderline case of [7].…”

“…For the area integrand f (x, z) = 1 + |z| 2 in codimension N = 1, for instance, Miranda's boundary continuity result [28,29] yields full uniqueness in case of a continuous boundary datum on a Lipschitz domain, while the examples of Santi [32] and Baldo & Modica [6] show that uniqueness up to constants is optimal for general data. For a detailed discussion of such non-uniqueness phenomena, we refer also to [7] and [19, Chapter V.2].…”