# Minimization of the square of the gradient in a volume

1. Feb 21, 2008

### siclar

1. The problem statement, all variables and given/known data
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.

2. Relevant 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 $$\frac{\delta f}{\delta y}-\frac{d}{dx}(\frac{\delta f}{\delta y'})=0$$

3. The attempt at a solution
I have no idea how to apply this minimization principle with multiple variables.

2. Feb 22, 2008

### EnumaElish

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