Existence of quasi-static equilibrium solutions in the large-strain regime

Establish the existence of solutions y to the quasi-static equilibrium boundary value problem -div(P)=0 in Ω with y=f on Γ_D and Pn=g on Γ_N, where P(x)=det(G(x)) D_pW(x, ∇y(x) G(x)^{-1}) G(x)^{-T}, in the large-strain regime when the growth tensor G forces large elastic strains; in particular, show existence even in function spaces larger than the small-strain setting considered.

Background

The paper establishes local-in-time well-posedness in a small-strain regime by exploiting proximity to the identity and fixed-point arguments. In contrast, when growth induces large strains, standard tools fail and the linearization may not be an isomorphism, obstructing existence theory.

The authors point out that even the existence of solutions to the quasi-static momentum balance in the large-strain regime is not known, highlighting a significant gap between small- and large-strain analyses for growth-induced deformations.

References

However, one of the main challenges in the large strain regime lies in the fact that a solution of equation for total deformation, even on a much larger space, is unknown to exist if the growth tensor forces the strains to be large.

— Morphoelastic Growth in the Presence of Nutrients at Small Strains  (2604.01812 - Abels et al., 2 Apr 2026) in Remark, Section "Local Existence of Solutions"