Functional and variational aspects of nonlocal operators associated with linear PDEs

We introduce a general difference quotient representation for non-local operators associated with a first-order linear operator. We establish new local to non-local estimates and strong localization principles in various topologies, which fully generalize those known for gradients. We also establish the invariance of quasiconvexity within the proposed local-nonlocal setting, resulting in a definition of $\mathcal A$-quasiconvexity that does not depend on derivatives. Applications to the fine properties of anisotropic gradient measures are further discussed.Comment: 32 pages (new critical estimates and applications with respect to the previous version

A Bourgain-Brezis-Mironescu representation for functions with bounded deformation

We establish a difference quotient integral representation for symmetric gradient semi-norms in $W^{1,p}(\Omega)$, $LD(\Omega)$ and $BD(\Omega)$. The representation, which is inspired by the formulas for the $W^{1,p}(\Omega)$ semi-norm introduced by Bourgain, Brezis and Mironescu and for the total variation semi-norm of $BV(\Omega)$ by Davila, provides a criterion for the $L^p$ and total-variation boundedness of symmetric gradients that does not require the understanding of distributional derivatives.Comment: 21 page

An elementary approach to the homological properties of constant-rank operators

We give a simple and constructive extension of Rai\c{t}\u{a}'s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Rai\c{t}\u{a}, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincar\'e lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.Comment: v3 22 pages, we added an an observation about the homology associated with operators acting on periodic maps, comments still welcome