We study the relaxation of integral functionals defined on orientation-preserving planar Sobolev homeomorphisms in W^{1,p}, 1 < p < ∞. Under natural assumptions on the stored energy density, the relaxed functional is again an integral functional. The assumptions are p-growth and p-coercivity, asymptotically p-homogeneous behavior at infinity, and agreement with a polyconvex function on singular matrices. Its integrand is the homeomorphic quasiconvex envelope of the original density, obtained by testing quasiconvexity with homeomorphisms only.
A key ingredient, of independent interest, is a boundary-modification result. Any sequence of planar homeomorphisms converging weakly to the identity can be replaced by finitely piecewise affine homeomorphisms that equal the identity on the boundary, have asymptotically the same gradients in measure, and exceed the original energy only by a contribution from a vanishing boundary layer. The lower bound combines this construction with a blow-up argument based on generalized Young measures, while the recovery sequence is built from piecewise affine approximations of Sobolev homeomorphisms.
Joint work with D. Campbell (Prague), S. Krömer (Prague) and E. Radici (L’Aquila).