On boundary value problems of nonlinear elastostatics (Q2565070)

The author proves that a parametrized mixed boundary value problem of nonlinear elastostatics has a unique solution in a neighborhood of a reference configuration, if the associated linear problem is pointwise stable and the load perturbations are small. The proof is based, among others, on the following existence theorem for the associated linear problem: Let \(1<p< \infty\), \(s> 1/p+1\), \(0\leq \alpha<1\), and let \({\mathbf H}^{s,p} (\Omega, {\mathbf R}^3)\) and \({\mathbf B}_\alpha^{s-1-1/p,p} (\partial\Omega, {\mathbf R}^3)\) denote the Sobolev and Besov spaces, respectively. If the elasticity tensor \({\mathbf C}\) is symmetric and pointwise stable, then for any \({\mathbf f} \in{\mathbf H}^{s-2,p} (\Omega, {\mathbf R}^3)\) and any \({\mathbf g} \in{\mathbf B}_\alpha^{s-1-1/p,p} (\partial\Omega, {\mathbf R}^3)\), the problem \(\text{div} ({\mathbf C} [\nabla {\mathbf u}]) ={\mathbf f}\) on \(\Omega\), \(\alpha ({\mathbf C} [\nabla {\mathbf u}] {\mathbf n}) +(1-\alpha) {\mathbf u}= {\mathbf g}\) on \(\partial\Omega\), has a solution \({\mathbf u} \in {\mathbf H}^{s,p} (\Omega, {\mathbf R}^3)\).