Variational problems of nonlinear elasticity theory in certain classes of mappings with finite distortion

Abstract

We study the problem of minimizing the functional $$I(\phi )=\underset{\Omega }{\int }W(x,D\phi )dx$$ on a new class of mappings. We relax summability conditions for admissible deformations to $\phi \in W_n^1(\Omega )$ and growth conditions on the integrand $W(x,F)$. To compensate for that, we impose the finite distortion condition and the condition $\frac{|D\phi (x){|}^n}{J(x,\phi )}\le M(x)\in L_s(\Omega )$, $s>n-1$, on the characteristic of distortion. On assuming that the integrand $W(x,F)$ is polyconvex and coercive, we obtain an~existence theorem for the problem of minimizing the functional $I(\phi )$ on a new family of admissible deformations. KEYWORDS: functional minimization problem, nonlinear elasticity, mapping with finite distortion, polyconvexity.