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

Abstract

We study the problem of minimizing the functional $$ I(\varphi)=\int\limits_{\Omega} W(x,D\varphi)\,dx $$ on a new class of mappings. We relax summability conditions for admissible deformations to $\varphi\in W^1_n(\Omega)$ and growth conditions on the integrand $W(x,F)$. To compensate for that, we impose the finite distortion condition and the condition $\frac{|D\varphi(x)|^n}{J(x,\varphi)} \leq M(x) \in L_{s}(\Omega)$, $s>n-1$, 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 $I(\varphi)$ on a new family of admissible deformations. KEYWORDS: functional minimization problem, nonlinear elasticity, mapping with finite distortion, polyconvexity.