Relaxation of p-Growth Integral Functionals Under Space-Dependent Differential Constraints

Abstract

A representation formula for the relaxation of integral energiesis obtained, where f satisfies p-growth assumptions, 1 < p < +∞, and the fields v are subjected to space-dependent first order linear differential constraints in the framework of A -quasiconvexity with variable coefficients.

Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints

“…The advantage of having a general and systematic approach to these problems turned out to be very useful both in theory [27,29] and applications [68,69]. Other recent works concerning the A-free framework include [4,11,41] and we also refer the reader to the very recent papers [3,21,25,80].…”

Quasiconvexity, Null Lagrangians, and Hardy Space Integrability Under Constant Rank Constraints

“…In the last four decades, the theory was developed much further, having found applications in Continuum Mechanics [31][32][33], Homogenization [11,15,73,74] and Nonlinear Analysis [4,29,42,58,80]. We also refer the reader to the recent papers [3,21,25,85].…”

Homogenization of integral energies under periodically oscillating differential constraints

“…The characterization of A -quasiconvexity has been extended to operators with variable coefficients in [70]. We refer to [38,39] for homogenization results in this purview and to [40] for a corresponding relaxation formula. Applications to the theory of compressible Euler systems, as well as to adaptive image processing and to data-driven finite elasticity were the subject of [25], [41], and [26], respectively.…”