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On derivation of Euler-Lagrange equations for incompressible energy-minimizers. (English) Zbl 1183.35264
The main result of the paper proves that a weak solution \(q\) of the problem \( \nabla q=\text{div}f\) posed in \(\mathcal{D}^{\prime }(U,\mathbb{R}^{n})\) belongs to a Hardy space \(h^{r}(V)\) for any \(V\subset \subset U\). Here \(U\) is a bounded, open, connected and Lipschitz domain of \(\mathbb{R}^{n}\), \( n\geq 2\) and \(f\) is a second-order tensor whose components belong to \( h^{r}(U)\). Moreover \(q\) can be locally represented as the sum of singular integrals which involve Calderon-Zygmund kernels and the components of \(f\). The main tools of this result are the use of mollifiers, \(h^{r}\)-estimates on different singular integrals and the properties of Calderon-Zygmund kernels. The last parts of the paper present applications of this general result to special cases. The authors first consider an incompressible Mooney-Rivlin bulk energy. They here prove the existence of a hydrostatic pressure \(q\) as solution of the problem \(\nabla q=\text{div}\widetilde{ \sigma }\) where \(\widetilde{\sigma }\) is the Cauchy-Green strain tensor associated to the displacement field \(\mathbf{u}\) which is a continuous and injective local minimizer of the bulk energy. They derive the Euler-Lagrange equations for the solution \((\mathbf{u},p)\) with \(p=q\circ \mathbf{u}\). The last result concerns a partial regularity of area-preserving minimizers.

MSC:
35Q74 PDEs in connection with mechanics of deformable solids
35B65 Smoothness and regularity of solutions to PDEs
42B35 Function spaces arising in harmonic analysis
49K20 Optimality conditions for problems involving partial differential equations
74B20 Nonlinear elasticity
74G65 Energy minimization in equilibrium problems in solid mechanics
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