Functional

A functional is a function whose inputs are themselves functions, curves, or other elements of a function space.

A quadratic functional on a real vector space combines a quadratic form $B(v,v)$, a linear functional $\ell (v)$ and a constant. If $B$ is a symmetric bounded bilinear form, its first variation is $DI(v)[w]=2B(v,w)-2\ell (w)$. For a strictly positive $B$, an existing stationary point is the unique minimizer by the quadratic variational principle for a symmetric positive operator.

The epigraph is $\{(u,t):E(u)\le t\}$. It is a convex set exactly when $E$ is a convex function, and is a sequentially closed set in the product topology exactly when $E$ is sequentially lower semicontinuous.

The functional derivative is defined by the first variation

Along a differentiable path of fields $\varphi (x,t)$, a differentiable functional obeysIntegrating in time expresses the functional change as the line integral of its functional derivative along the path.

An energy functional assigns an integral energy to a function. Minimization of a suitable energy is the basis of the Dirichlet principle. For the modified Helmholtz equation, a natural example is

The p-energy of a Sobolev function isIts stationary points under fixed boundary data are weak solutions of the p-Laplacian equation.

A Lagrangian is a function or density whose stationary integral encodes an extremization problem. In Lagrangian mechanics its action gives the equations of motion; in geometry it describes geodesics and minimal surfaces.

For an objective $f(x)$ with equality constraints $g_i(x)=0$, the Lagrangian function is

A variation of a function $y$ is a one-parameter family $y+\epsilon \eta$, where $\eta$ satisfies the required boundary conditions.

The first variation of a functional $L$ at $y$ in the direction $\eta$ is

If a continuous function $f$ satisfiesfor every smooth compactly supported test function $\eta$, then $f=0$ on $(a,b)$.