Minimization of the square of the gradient in a volume

[siclar]

Homework Statement

Find an expression involving the function $$\phi(x_1, x_2, x_3)$$ that has a minimum average value of the square of its gradient within a certain volume V of space.

Homework Equations

We are studying functionals, though so far it has only been of one variable. We're considering $$J[y]=\int_{x_1}^{x_2}f(y, y'; x)dx$$ where $$y$$ is a function of x. This functional is minimized when f satisfies $$\frac{\delta f}{\delta y}-\frac{d}{dx}(\frac{\delta f}{\delta y'})=0$$

The Attempt at a Solution

I have no idea how to apply this minimization principle with multiple variables.

 

[EnumaElish]

A multivariate function is minimized when its first-order partial derivatives with respect to all of its arguments are simultaneously zero.