The nonlinear membrane energy: variational derivation under the constraint ``\(\det \nabla u\neq 0\)''
Let \( \Sigma_{\varepsilon} \subset \mathbb R^3\) be of the form \(\Sigma_{\varepsilon}=\Sigma \times (-\frac{1}{\varepsilon}, \frac{1}{\varepsilon})\) where \(\Sigma \subset \mathbb R^2\) is Lipschitz open and bounded, and define the integral functionals of the form \[ E_{\varepsilon}(u)= \frac{1}{\varepsilon} \int_{\Sigma_{\varepsilon}} W(Du)\,dx \] where \(W(z)\) be the continuous stored-energy function such that \(W(z)=+\infty\) if and only if \(\det z=0\) and is coercive when \(\det z \geq \delta\). The authors prove that as \(\varepsilon \rightarrow 0\) the three-dimensional free energy \(E_{\varepsilon}\) converges in a variational sense, which is a variant of De Giorgi \(\Gamma\)-convergence, to the two-dimensional free energy \[ E_{\text{mem}}(u)= \int_{\Sigma} W_{\text{mem}}(Du)\,dx \] and give a representation formula for \(W_{0}\). The result permits to obtain the nonlinear membrane energy \(E_{\text{mem}}\) under the constraint \(\det Du \neq 0\).
scientific article; zbMATH DE number 5129882