American Institute of Mathematical Sciences

$\int$_Ω$\l(u(x),\rho_1(x,u(x)) $∇$u(x))\ \ \rho_2(x,u(x))\ dx \ \ \ on\ \ \ $W^1,1_u∂(Ω),(*)

where Ω$\subset\R^d$ is open bounded, $u:$Ω$\toR$ is in the Sobolev space u_∂($*$)+W^1,1₀(Ω), with boundary data $u_$_∂$(\cdot)\in$W^1,1(Ω)$\cap C^{0}$(Ω); and $\l:R$Χ$R^d\to[0,\infty]$ is superlinear $L\oxB$-measurable with $\rho_1(\cdot,\cdot),\rho_2(\cdot,\cdot)\in C^{0}($ΩΧ$R)$ both $>0$.
   One main feature of our result is the unusually weak assumption on the lagrangian: l^**$(\cdot,\cdot)$ only has to be $lsc$ at $(\cdot,0)$, i.e. at zero gradient. Here l^**$(s,\cdot)$ denotes the convex-closed hull of $\l(s,\cdot)$. We also treat the nonconvex case $\l(\cdot,\cdot)\ne$l^**$(\cdot,\cdot)$, whenever a well-behaved relaxed minimizer is a priori known.
   Another main feature is that $\l(s,\xi)=\infty$ is freely allowed, even at zero gradient, so that (*) may be seen as the variational reformulation of optimal control problems involving implicit first-order nonsmooth scalar partial differential inclusions under state and gradient pointwise constraints.
   The general case $\int$_Ω$L(x,u(x),$∇$u(x))$ is also treated, though with less natural hypotheses, but still allowing $L(x,\cdot,\xi)$ non-$lsc$ for $\xi\ne0$.

$\int_{a}^{b}L( x( t) ,x^'( t)) d\,t,\text{ \ }x\( \cdot) \in W^{1,1}((a,b) ,\mathbb{R}^{n}) ,\text{ \ }x(a)=A\,x(b) =B\ \ $(*)

whose velocities are not a.e. constrained by the domain boundary.
   Thus we do not ask ( as preceding results do) the free-velocity times

$ T_{f ree}:=\{ t\in[ a,b] :y^'( t) \in $int$\text{ }dom\ L( y\( t) ,\cdot) \} $

to have "full measure"; on the contrary, "positive measure" of $T_{f ree}$ suffices here to guarantee the above necessary condition.
   One main feature of our result is that $L( S,\xi) =\infty$ freely allowed, hence the domains $dom$$L( S,\cdot) $ may be e.g. compact and (*) can be seen as the variational reformulation of general state-and-velocity constrained optimal control problems.
   Another main feature is the clean generality of our assumptions on $ L( \cdot) :$ any Borel-measurable function $L:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow[ 0,\infty] $ having $L( \cdot,0) $ $lsc$ and $L( S,\cdot) $ convex $lsc$ $\forall\,S.$
   The nonconvex case is also considered, for $L( S,\cdot) $ almost convex lsc $\forall\,S.$