Boundary effects and weak$^*$ lower semicontinuity for signed integral functionals on $\mathrm{BV}$

TL;DR

This paper characterizes the conditions under which integral functionals are lower semicontinuous with respect to weak* convergence in BV spaces, especially considering boundary effects and integrands with negative parts of linear growth.

Contribution

It introduces a new boundary condition called quasi-sublinear growth from below, extending previous results on lower semicontinuity in BV spaces.

Findings

Established a characterization of lower semicontinuity for integral functionals in BV.

Identified the role of boundary shape and integrand behavior in lower semicontinuity.

Extended recent results by Kristensen and Rindler to more general boundary conditions.

Abstract

We characterize lower semicontinuity of integral functionals with respect to weak$^*$ convergence in $\mathrm{BV}$, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition -- quasi-sublinear growth from below at points of the boundary -- which compensates for possible concentration effects generated by the sequence. Our work extends some recent results by J. Kristensen and F. Rindler (Arch. Rat. Mech. Anal. 197 (2010), 539--598 and Calc. Var. 37 (2010), 29--62).