4.5 Article

Finding nicer mappings with lower or equal energy in nonlinear elasticity

Journal

Publisher

ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2020.123918

Keywords

Mapping; Lower; Energy; Nonlinear; Elasticity; Maximum

Funding

  1. research project Integro-differential Equations and nonlocal Problems - Fondazione di Sardegna
  2. grant PRIN [PRIN-2017AYM8XW]

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

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