Existence of minimizers and lower semicontinuity of integral functionals in the vectorial case
The main existence result of the paper concerns the Monge-Ampère boundary problem \[ \begin{cases}\text{det }D^2u(x)=f(x),\qquad\text{a.e. in }\Omega,\\ Du(x)=x,\qquad\text{on }\partial\Omega,\end{cases} \] where \(\Omega\) is a bounded open set of \(\mathbb R^n\) with \(|\partial\Omega|=0\), and \(f\) is a positive continuous function on \(\overline\Omega\) such that \(\int_\Omega f=|\Omega|\). The authors prove the existence of a solution \(u\in W^{2,p}(\Omega)\cap C^1(\Omega)\) for every \(1\leq p<\infty\) which minimizes the underlying functional \(J\) in a suitable set \(W\), extending a previous result due to \textit{B. Dacorogna} and \textit{J. Moser} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 1, 1-26 (1990; Zbl 0707.35041)]. With the same technique other boundary problems are solved, and also criteria for lower semicontinuity of the functionals \(\int_\Omega F(u(x),\text{det }Du(x))\) are given under assumptions usually verified in nonlinear elasticity.
scientific article; zbMATH DE number 1395894
scientific article