# Calculus of Variations Boundary Terms in Higher Dimensions

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?

• 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? – Botond May 16 at 0:35
• @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. – user7530 May 16 at 1:27
• What is the Lagrangian of the system? – Qmechanic May 17 at 18:40