On the Lipschitz Regularity for Minima of Functionals Depending on $x$, $u$, and $ abla{u}$ under the Bounded Slope Condition
Flavia Giannetti, Giulia Treu · SIAM Journal on Control and Optimization · 2022
We prove the existence of a global Lipschitz minimizer of functionals of the form $\mathcal I(u)=\int_\Omega f( abla u(x))+g(x,u(x))\,dx$, $u\in\phi+W^{1,1}_0(\Omega)$, assuming that $\phi$ satisfies the bounded slope condition (BSC). Our assumptions on the Lagrangian allow the function $f$ to be strongly degenerate.