The role of intrinsic distances in the relaxation of L∞-functionals

We consider a supremal functional of the form F(u) = ess sup x∈Ω f(x,Du(x)) where Ω ⊆ RN is a regular bounded open set, u ∈ W1,∞(Ω) and f is a Borel function. Assuming that the intrinsic distances dλ F (x, y) := sup { u(x) − u(y) : F(u) ≤ λ } are locally equivalent to the euclidean one for every λ > infW1,∞(Ω) F, we give a description of the sublevel sets of the weak∗-lower semicontinuous envelope of F in terms of the sub-level sets of the difference quotient functionals RdλF(u) := supx̸=yu(x)−u(y)dλF (x,y) . As a consequence we prove that the relaxed functional of positive 1-homogeneous supremal functionals coincides with Rd1F. Moreover, for a more general supremal functional F (a priori non coercive), we prove that the sublevel sets of its relaxed functionals with respect to the weak∗ topology, the weak∗ convergence and the uniform convergence are convex. The proof of these results relies both on a deep analysis of the intrinsic distances associated to F and on a careful use of variational tools such as Γ-convergence.