DCM
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
@mnmn1993: just out of curiosity... where does your $E$ come from? In particular, did you start with the functional or the PDE?
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
I nice analogy here is 'plotting the graph of a function'. $E'(u)$ is a linear function(al) on $H^1(B)$; we learn about its `shape' behaviour by plugging in values and seeing what we get. The behaviour of $E'(u)$ then tells us things about $u$.
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
I think the slickest way to get the interior equation and the boundary equation alone is by choosing test functions $v$ which satisfy a suitable PDE of their own (a PDE designed to make one of the two integrals vanish).
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
In particular, I'd be inclined to see what you get by using Green's identity (thought of in a distributional sense, at least initially) in the equation in your third display, and seeing what drops out of that for different choices of $v$.
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
I think the general pattern you follow is sound. Assuming $E:H^1(B)\to \mathbb{R}$ is differentiable, a necessary condition for $u\in H^1(B)$ to be a minimiser is that $E'(u)v = 0$ for all $v\in H^1(B)$. Choosing $v\in H^1_0(B)$ gives you your "EL" equation, for example. The key thing is that $u$ has to satisfy the $E'(u)v = 0$ whatever $v\in H^1(B)$ you choose, which is the main mechanism through which you're able to deduce things about its behaviour.
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
Just to make sure I understand correctly: is the goal here to show that the minimising $u_0\in H^1(B)$ is a $H^2(B)$ function satisfying your Neumann problem?
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Construction of elliptic equation with Neumann boundary condition from a minimization problem
From the context, it looks like your "$E:\mathbb{R}\to \mathbb{R}$" (third line) should be "$E:H^1(B)\to \mathbb{R}$".
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Rellich Embedding Theorem for the $2$-Sphere
Chapter X of "Seminar on the Atiyah-Singer index theorem" (Palais) seems like another useful reference for this sort of thing, especially if you're interested in Sobolev spaces associated with vector bundles. It's a very nice book, in any case.
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Rellich Embedding Theorem for the $2$-Sphere
Technical geometric details aside, the crucial point here is that one can find parameterisations $U_i\to X_i\subset X$ such that pullback gives equivalence between the 'upstairs' norm $\Vert .\Vert_{H^1(X_i)}$ is equivalent to the `downstairs' norm $\Vert .\Vert_{H^1(U_i)}$ for compactly supported smooth functions. Once you have this, I'm not sure there's any real difference between working on a compact manifold and working in a nice bounded domain in $\mathbb{R}^n$