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I want to find the function $y(x)$ that minimizes a functional $J(y)$ which is not one single integral, but something more general, for example a product of integrals such as:

$J(y)=\int L_{1}(x,y,y')dx \int L_{2}(x,y,y')dx$

What would be the approach in that case?

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    $\begingroup$ $J(y)$ still defines a real valued functional. Therefore, you could try to compute the Gateaux derivative (first variation) of the functional in the direction of an admissible $\phi \in C_c^{\infty}$ (or something suitable). Depending on how familiar you are with the calculus of variations, the direct method of the calculus of variations might still work, as the general procedure does not require the functional to have any specific form. Just try the "usual steps": Find compactness properties for the minizing sequence and hope for lower semicontinuity. $\endgroup$ – F. Conrad Apr 3 at 1:11

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