\(T\)-minima: an approach to minimization problems in \(L^1\)
The author considers functionals of the form \[ J(v)=\int_{\Omega}j(x,\nabla v) dx - \int_{\Omega}f(x)v(x) dx, \] where \(j:\Omega\times\mathbb{R}^N \to\mathbb{R}\) is assumed to satisfy the standard hypotheses for integrands in the calculus of variations so that, if \(f\in L^{p'}(\Omega)\), then \(J\) possesses a unique minimizer in \(W^{1,p}_0(\Omega)\), where \(p,p'>1\) are conjugate exponents. If, however, \(f\in L^1(\Omega)\), then the functional \(J\) in general fails to possess a minimizer. The author now defines the \(T\)-minimum of \(J\) which involves certain truncations of the admissible functions. Under the standard hypotheses on \(j\), he proves that \(J\) has precisely one \(T\)-minimum. The continuous dependence of \(T\)-minima under weak-\(L^1\) perturbations is also studied as well as the relationship to entropy solutions of the Dirichlet problem.
scientific article; zbMATH DE number 1884307