Pointwise and integral constraints have different regularity tests

The multiplier rules of this page apply to equality constraints given by a $C^1$ map with surjective derivative and produce a multiplier equation (The Lagrange multiplier rule for finitely many regular constraints, The Lagrange multiplier rule for one regular constraint in Hilbert space), while an obstacle constraint $u\ge \psi$ is a closed convex inequality constraint that does not by itself provide a differentiable multiplier field: its first-order information is the variational inequality $a(u,v-u)\ge F(v-u)$ (The closed convex obstacle set and the obstacle variational inequality), and complementarity is expressed through the reaction distribution, not through a pointwise product (Obstacle complementarity in distribution form).

Two different regularity tests are therefore in force. The equality rule needs surjectivity of the constraint derivative at the extremum; when that test fails the multiplier equation can fail outright, as The degenerate constraint $x^2+y^2=0$ defeats the multiplier conclusion records for the degenerate constraint $x^2+y^2=0$. The obstacle set has no derivative to test and instead needs regularity of the reaction if one wants more than the distributional inequality: the function version is stated with an $L^2$ reaction density and continuous representatives, while the measure version uses a Radon representation and continuous representatives (Obstacle complementarity in distribution form, The obstacle reaction is supported on the contact set under measure regularity). Minimality alone supplies neither an $L^2$ density nor continuous representatives; the conditional measure formulation does not assert that a nonnegative distribution can fail to admit a Radon representation. In particular the smooth finite-dimensional Lagrange multiplier theorem must not be applied to the obstacle set.