Integral functional with a Lagrangian density

Remarks

In one variable, with $n = m = 1$ and $\Omega = (a, b)$, the functional reads $J(y) = \int_a^b L(x, y(x), y'(x)) \, dx$. Example: $L(x, z, p) = \frac{1}{2} \abs{p}^2$ with $m = 1$ gives the Dirichlet energy $J(u) = \frac{1}{2} \int_\Omega \abs{\nabla u}^2 \, dx$. It is an integral functional because this $L$ is continuous on all of $\overline{\Omega} \times \R \times \R^n$. Non-example: on $\Omega = (0, 1)$, the expression $L(x, z, p) = 1/x$ is not defined at $x = 0$ and has no continuous extension to $[0, 1] \times \R \times \R$, and indeed $\int_0^1 dx / x = \infty$ for every $y$. The same name is used for Lagrangians with higher derivatives, as in the Euler–Poisson equation, and for $u$ in Sobolev spaces under growth conditions on $L$.