Constrained variations and multipliers

Work in a Banach space of functions, for example a fixed-trace affine subset of $H^1(a,b)$ with homogeneous perturbation space $H_0^1$. Let the objective and a scalar equality constraint be continuously Fréchet differentiable nearby. If a differentiable feasible curve through the candidate has tangent $h$, differentiating the equality gives $DG[u](h)=0$. Thus feasible tangents lie in the kernel of the constraint derivative. A tangent condition alone does not make the straight line $u+\varepsilon h$ exactly feasible for a nonlinear constraint.

For an area constraint $G[u]=\int u$, the constraint is affine, so zero-integral directions preserve it exactly along straight lines. For a normalization $G[u]=\int u^2$, the tangent condition is $\int uh=0$, but the second-order term $\varepsilon^2\int h^2$ changes the norm along the uncorrected line. One can instead normalize a nearby perturbed function to construct a feasible curve. Keeping exact feasibility separate from its linearization prevents incorrect second-variation calculations on constrained problems.

For finitely many equality constraints in a Banach space, a standard multiplier theorem applies when the objective and constraints are continuously Fréchet differentiable and the derivative of the constraint map onto the finite-dimensional constraint space is surjective. At a constrained local minimum, the objective derivative is then a linear combination of the constraint derivatives. For one scalar constraint this qualification means its derivative is not the zero functional. The theorem follows from the local level-set structure supplied by the implicit-function theorem.

Choose the sign convention $J-\lambda(G-m)$ for the augmented functional. If both quantities are integrals, apply the Euler–Lagrange calculation to the augmented density $F-\lambda g$. The multiplier is one constant for one global scalar equality, because the equality is a single number. A pointwise constraint instead involves a field of restrictions and can require a function-valued multiplier in an appropriate dual space. These are different problems; replacing an integral constraint by a pointwise equation radically changes the admissible set.

Constraint qualifications cannot be omitted. In the finite-dimensional example minimizing a linear function subject to a squared variable equal to zero, the feasible set contains only zero, yet the constraint derivative there is zero while the objective derivative is not. No normal multiplier represents the objective derivative. Similar degeneracies can occur in function spaces. A failed qualification does not mean there is no constrained minimizer; it means this particular normal-multiplier conclusion is unavailable without another argument.

For convex quadratic energies with an affine integral constraint, a complete comparison is often simple. Once the candidate satisfies the multiplier equation, its first-order cross term against any feasible difference is the multiplier times the zero constraint difference. What remains is a positive quadratic energy. This proves uniqueness and global minimality in the specified class. For nonconvex functionals or nonlinear constraints, second-order tangent corrections and competing branches matter, so solving an augmented differential equation alone is not enough.