On a new class of elastic deformations not allowing for cavitation. (English) Zbl 0863.49002

Müller, S.; Qi, Tang; Yan, B. S.

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11, No. 2, 217-243 (1994).

The divergence identity \[ \partial_j\{(g^i\circ u)(\text{adj }Du)^j_i\}= (\text{div }g)\circ\text{det }Du \] is proved under weak assumptions. Also, an integrability result for \(\text{det }Du\) is proved. As an application, a degree formula is proved in the class \(A_{p,q}(\Omega)\). As a further application, the existence of an absolutely continuous minimizer of \[ I(u)=\int_\Omega W(x,u,Du)dx \] is proved if \(W\) is polyconvex and satisfies the inequality \(W(x,u,F)\geq a(|F|^p+|\text{adj }F|^q)\) with \(p\geq 2\) and \(q\geq 3/2\).

Reviewer: M.Brokate (Kaiserslautern)

MSC:

49J10	Existence theories for free problems in two or more independent variables
26B10	Implicit function theorems, Jacobians, transformations with several variables
74B20	Nonlinear elasticity

Keywords:

nonlinear elasticity; divergence identity; degree formula; absolutely continuous minimizer

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