Special Session 32: 

Lipschitz regularity of the minimizers for non autonomous problems with slow growth of the calculus of variations

Carlo Mariconda
University of Padua
Italy
Co-Author(s):    Piernicola Bettiol
Abstract:
We consider the Lipschitz regularity of the minimizers to the constrained problem of the calculus of variations: \[\min\int_a^bL(t, x(t), x`(t))\, dt,\,x\in W^{1,1}([a,b]),\] \[x(a)=A, x(b)=B, x(t)\in \Sigma\subset R^n.\] A celebrated result by F. Clarke and R. Vinter asserts the Lipschitz regularity of the minimizers when the Lagrangian $L$ is \emph{autonomous}, locally Lipschitz and convex in the velocity variable. We prove that the result holds true for \emph{non autonomous} lagrangians by assuming merely that the Lagrangian is \emph{Borel}, locally Lipschitz (just) in $t$, under a \emph{weak growth condition} that is slower than superlinearity. The proof relies upon a new extension of the Du Bois-Reymond necessary condition, obtained via Clarke`s Extended Maximum Principle.