| 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. |
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