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Minimization of the square of the gradient in a volume

  1. Feb 21, 2008 #1
    1. The problem statement, all variables and given/known data
    Find an expression involving the function [tex]\phi(x_1, x_2, x_3)[/tex] 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 [tex]J[y]=\int_{x_1}^{x_2}f(y, y'; x)dx[/tex] where [tex]y[/tex] is a function of x. This functional is minimized when f satisfies [tex]\frac{\delta f}{\delta y}-\frac{d}{dx}(\frac{\delta f}{\delta y'})=0[/tex]

    3. The attempt at a solution
    I have no idea how to apply this minimization principle with multiple variables.
  2. jcsd
  3. Feb 22, 2008 #2


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