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Existence of solution to Laplace equation given graph form and fractional Sobolev data

Given that $\Omega$ is a smooth Riemannian manifold such that $\bar \Omega$ is compact and the boundary $\partial \Omega$ is smooth, there exists a unique weak solution $\varphi \in H^1(\Omega)$ to ...
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Laplace equation from a functional solved on a disk - arising from a functional equation

The Lagrange multiplier condition for extremizing this functional subject to these constraints is $\nabla^2 z = -\lambda z$, as you found. I can derive this from scratch if desired but it seems ...
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Relation between eigenvalue and representation of compact Lie group

I don't know the precise details but this is how it should go in broad strokes. If $G$ is a compact semisimple Lie group it can be equipped with a canonical Riemannian metric given by translating the ...
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Evaluation of $~\Re\{ \exp(x+iy) \}~$ to proceed to use laplacian operator.

$e^{x+iy}=e^x\,e^{iy}$ $e^{iy}=(\cos y+ i\sin y)$ Hence $e^{x+iy}=(\cos y+ i\sin y)e^x$ Can you get it from here?
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