Relaxation of variational problems in the space of functions with bounded $\mathcal{B}$-variation: interaction with measures and lack of concentration phenomena

TL;DR

This paper establishes an integral representation for variational functionals in the space of functions with bounded $\

Contribution

It introduces a new integral representation for $BV^{\\mathcal{B}}$ spaces involving higher-order differential operators and analyzes concentration phenomena and measure interactions.

Findings

Explicit representation of relaxed energies with linear growth.

Identification of concentration phenomena depending on dimension and operator order.

Extension to Sobolev-type and variable exponent spaces.

Abstract

We prove an integral representation result for variational functionals in the space $BV^{\mathcal{B}}$ of functions with bounded $\mathcal{B}$-variation where $\mathcal{B}$ denotes a $k$-th order, $\mathbb{C}$-elliptic, linear homogeneous differential operator. This result has been used as a key tool to get an explicit representation of relaxed energies with linear growth which lead to limiting generic measures. According to the space dimension and the order of the operator, concentration phenomena appear and an explicit interaction is featured. These results are complemented also with Sobolev-type counterparts. As a further application, a lower semicontinuity result in the space of fields with $p(\cdot)$-bounded $\mathcal{B}$-variation has also been obtained.