We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev–Slobodeckiĭ seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.

For smooth functions ${{a}_{1}}\,,\,{{a}_{2}}\,,\,{{a}_{3}}\,,\,{{a}_{4}}\,$ on a quaternion Heisenberg group, we characterize the existence of solutions of the partial differential operator system ${{X}_{1}}f\,=\,{{a}_{1}},\,{{X}_{2}}f=\,{{a}_{2}},\,{{X}_{3}}f\,=\,{{a}_{3}},\,\text{and}\,{{X}_{4}}f\,=\,{{a}_{4}}$. In addition, a formula for the solution function $f$ is deduced, assuming solvability of the system.

We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.