Regularity of solutions for some variational problems subject to convexity constraint (Q2708176)

The authors study the regularity of minimizers of problems of the form NEWLINENEWLINENEWLINE`Minimize NEWLINE\[NEWLINE\int_\Omega A(x,u) \nabla u(x)\cdot \nabla u(x) dx + \langle f,u \rangleNEWLINE\]NEWLINE in the class \({\mathcal C}\cap {\mathcal H}\), where \(\Omega\) is a bounded convex domain of \(R^N\), \(A:\Omega \times R \mapsto R^N\) is a continuous map of symmetric matrices satisfying an ellipticity condition, \(f=f^+ - f^-\), with \(f^+ \in L^N(\Omega)\) and \(f^-\) is a nonnegative measure, \({\mathcal C}\) is the set of convex functions on \(\overline \Omega\), \({\mathcal H}= u_0 + {\mathcal H}_0\), \({\mathcal H}_0\) is the closure of \(C_c^\infty(\Omega)\) for the norm \(\|v\|_{{\mathcal H}_0}= (\int_\Omega \alpha(x) |\nabla u|^2)^{1/2}\), \(\alpha \) is a positive weight'. NEWLINENEWLINENEWLINEIt is proved that if \(\Omega\) is \(C^{1}\) and \(u_0\) is \(C^{1}\), then any minimizer of the above problem is of class \(C^{1}\). An application to a model in economics is given.