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Variational approach to contact problems in nonlinear elasticity. (English) Zbl 1050.35015
The paper under review deals with some variational problems arising from 3-dimensional nonlinear elasticity, where the definition of the elastic body is restricted by a rigid obstacle. The total energy of the body deformed by the map \(u :\Omega-\mathbb{R}^3\) is given by
\[ E(u) = \int_\Omega W(x,Du)\,dx -\langle f,u\rangle, \]
where \(W\) is the stored energy density function and \(f\) represents the action of all external forces. Dirichlet boundary conditions on some part \(\Gamma_D\) of \(\partial\Omega\) are prescribed by imposing
\[ u = u_D\quad \text{on }\Gamma_D. \]
Finally, the presence of a rigid obstacle \(O\) is described by requiring that
\[ u(x)\in\overline {\mathbb{R}^3\setminus O} \quad\text{for all }x\in\Omega. \]
The variational problem is then \[ \min\{E(u) : u W^{1,p}(\Omega,\mathbb{R}^3),\;u = u_D \text{ on }\Gamma_D,\;u(x)\in \overline {\mathbb{R}^3\setminus O}\}. \] Under suitable assumptions on the data it is shown that an optimal solution exists. Moreover, the Euler-Lagrange equation is studied, and a rigorous interpretation of Lagrange multipliers is given.

35D05 Existence of generalized solutions of PDE (MSC2000)
74B20 Nonlinear elasticity
35Q72 Other PDE from mechanics (MSC2000)
49K20 Optimality conditions for problems involving partial differential equations
74M15 Contact in solid mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
35A15 Variational methods applied to PDEs
35J20 Variational methods for second-order elliptic equations
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