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In 2D I apply the calculus of variations to get an equation of the following form on a domain $\Omega\subset\mathbb{R}^2$, and I want to extract the Euler-Lagrange equations:

$$\int_{\Omega} f(p)\delta(p)\,dA + \int_{\partial \Omega} g(p)\delta(p)\,ds + \int_{\partial \Omega} h(p)\nabla \delta(p)\cdot \hat{n}\,ds = 0,$$ for $\delta(p)$ an arbitrary variation. Oftentimes it is given that $\delta$ or its derivative vanishes on the domain boundary, so that one of the boundary terms vanishes, but here $\delta$ is completely free.

Can I conclude that $f(p) = g(p) = h(p)=0$?

Clearly I can choose a bump function $\delta(p)$ at any interior point whose value and derivative vanishes on the boundary, so that $f(p)=0$. I'm less clear on whether I can independently also force $g(p)=0$ and $h(p)=0$. Intuitively, I ought to be able to take $\delta(p)$ to be a test function whose normal derivative is $1$ at a boundary point $p$, but whose normal derivative and value vanishes outside an arbitrarily small neighborhood?

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  • $\begingroup$ I'm only familiar with the volume integrals in calculus of variations, but won't you get a boundary condition if you apply partial integration on the last integral? $\endgroup$ – Botond May 16 at 0:35
  • $\begingroup$ @Botond if you move the derivative on the last term and apply Stokes's theorem, you get a volume integral involving $\nabla \delta$, which is counterproductive I think. $\endgroup$ – user7530 May 16 at 1:27
  • $\begingroup$ What is the Lagrangian of the system? $\endgroup$ – Qmechanic May 17 at 18:40

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