Lagrange multipliers and transport densities

Abstract

In this paper we consider a stationary variational inequality with nonconstant gradient constraint and we prove the existence of solution of a Lagrange multiplier, assuming that the bounded open not necessarily convex set O has a smooth boundary. If the gradient constraint g is sufficiently smooth and satisfies ?g 2 =0 and the source term belongs to L 8 (O), we are able to prove that the Lagrange multiplier belongs to L q (O), for 1 < q < 8, even in a very degenerate case. Fixing q=2, the result is still true if ?g 2 is bounded from above by a positive sufficiently small constant that depends on O, q, minO??g and maxO??g. Without the restriction on the sign of ?g 2 we are still able to find a Lagrange multiplier, now belonging to L 8 (O) ' . We also prove that if we consider the variational inequality with coercivity constant d and constraint g, then the family of solutions (? d ,u d ) d > 0 of our problem has a subsequence that converges weakly to (? 0 ,u 0 ), which solves the transport equation.