Finding nicer mappings with lower or equal energy in nonlinear elasticity

In the framework of nonlinear elasticity, we consider the model energy F(u) = integral(Omega)[vertical bar Du(x)vertical bar(p) + h(det Du(x))]dx, where u : Omega subset of R-n -> R-n with det Du > 0 and h : (0, +infinity) -> [0, +infinity) is convex; moreover h(t) blows up when t -> 0(+). We study the problem: fix the mapping u with finite energy F(u) and find a mapping v with the same boundary values, with det Dv > 0 and energy F(v) not higher than F(u), such that every component v(beta) of v enjoys a kind of maximum and minimum principle. Due to the constraint det Dv > 0, a truncation argument does not work. On the contrary, the constraint det Dv > 0 makes the problem easy when p >= n since it is known that every component is weakly monotone, so we can take v = u. In the present work we address the case 2 <= p < n and we give a positive answer if the blow up of h(t) is of logarithmic type. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

In the framework of nonlinear elasticity, the study focuses on a mapping problem under constraints, with positive results obtained when certain conditions are met.