Euler-Lagrange equations and Bellman's principle of optimality One method to optimize the integral 
$$\int_{\mathcal T} L(t,x,\dot{x}) \; dt $$
of a functional over a curve is the calculus of variations, which leads to ordinary differential equations: the Euler-Lagrange equation
$$-\frac{d}{dt} L_{\dot{x}}(t,x(t),\dot{x}(t)) + L_x(t,x(t),\dot{x}(t)) =0$$
 or its generalization, Pontryagin's maximum principle. An alternative is Bellman's optimality principle, which leads to Hamilton-Jacobi-Bellman partial differential equations. Each of the methods has advantages and disadvantages depending on the application, and there are numerous technical differences between them, but in the cases when both are applicable the answers are broadly similar. 
The calculus of variations can also be used to optimize a functional 
$$\int_{\mathcal X} L(x,u,p) \; dx $$
integrated over a multidimensional space. The resulting Euler-Lagrange equations $$-\frac{\partial}{\partial x} L_{p}(x,u(x),p(x)) + L_u(x,u(x),p(x))$$ are partial differential equations with the space coordinates as independent variables. Is an alternative approach using value functions, leading to optimality conditions along the lines of Bellman's optimality principle, known?
 A: A. multi-dimensional state, one-dimensional time
Multi-dimensional extensions $x\in\mathbb{R}^n$ of the one-dimensional Hamilton-Jacobi-Bellman equations have been considered in Consistency of a Simple Multidimensional Scheme for Hamilton-Jacobi-Bellman Equations (2005).

We present an approximation scheme for second-order
  Hamilton–Jacobi–Bellman equations arising in stochastic optimal
  control. The scheme is based on a Markov chain approximation method to solve the nonlinear partial differential equations that govern the optimization problem.
  The scheme can be readily implemented in any dimension. The consistency of the
  scheme is proved, which guarantees its convergence.


B. multi-dimensional state, multi-dimensional time
For extensions where both state and time are multi-dimensional, $x\in\mathbb{R}^n$, $t\in\mathbb{R}_+^m$, see Multitime linear-quadratic regulator problem based on curvilinear integral (2009) (and several more recent papers on the multi-time Bellman principle by Constantin Udriste and co-workers).

We introduce a multitime dynamic programming method based on multitime
  Hamilton- Jacobi-Bellman PDEs. These PDEs are equivalent to multitime
  Hamilton PDEs system and the multitime maximum principle. The
  optimal control is characterized means of a multitime variant of the
  Riccati PDE that may be viewed as a feedback law.

A: As I haven't noticed before, this one is a duplicate of the first answer:
Riemannian optimal control and this author's further works

Theorem (Multitime maximum principle). Suppose u∗(·) is an optimal
  solution of the control problem and x∗(·) is the corresponding
  optimal state. Then there exists a costate tensor...

And maybe this article, I don't have full access ATM:
The taxation principle and multi-time Hamilton-Jacobi equations

...Here, we propose an example of the contrary: every system of first
  order partial differential equations of a certain type can be solved
  by use of an economics principle. For the case of a single equation,
  our approach is in some sense dual to the usual optimal control
  method.

A: As far as I know a multidimensional version of Bellman's principle of optimality has not been found. The papers suggested in the answers above all refer to one-dimensional independent variables, or to cases that can be reduced, by introducing integrability assumptions, to optimization of a functional with a one-dimensional independent variable.
Finding a multivariable version of the dynamic programming method may be an open problem. Until it is solved, the classical Euler-Lagrange equations are the method we have.
