I take your point. However, it's possible to come up with similar examples where - instead of small oscillation - the objective function has lots of short flat sections (rather than a wiggly parabola like the one above, maybe think of a 'staircase with rounded corners').

Hint 1: your BCs can be written much more succinctly in terms of $\partial u/\partial n$. Hint 2: your $\int_{\partial\Omega}$ term wants to be part of $a(.,.)$.

Without having checked the details in the paper... I'd say that $U:\mathbb{R}^n\times [0,\infty)\to \mathbb{R}$ vanishes weakly as $y\to \infty$ if $\int_{\mathbb{R}^n} U(x,y)\phi(x)\mathrm{d}x\to 0$ as $y\to \infty$ for all $\phi$ in some class of test functions (e.g. $\mathscr{D}(\mathbb{R}^n)$).

I think the "icosahedron followed by recursive subdivision" strategy isn't very efficient in this respect - I think it's better to start with the icosahedron then do the subdivision 'in one shot' rather than iteratively. One thing I'm curious about: is your ultimate aim to sample from a uniform distribution on the sphere, or to produce a triangulation?

It seems like you're asking several questions here. Am I right to think your main one is along the lines of "for a fixed distance tolerance $\epsilon$, what subdivision strategy approximates the sphere to within tolerance with the fewest triangles?" (possibly with the addendum "if there are several, which one is most uniform?").

I should have said "Banach-ness, combined with being stable under differentiation" in my comment above ;)

My suspicion is that 'being stable under differentiation' isn't compatible with how fast the Fourier coefficients have to decay for $A\subset C^\infty$ (thinking about the $d=1$ case). Stability under multiplication isn't inconsistent with $A\subset C^\infty$ (some Beurling algebras and some 'Dales-Davie' algebras consist entirely of $C^\infty$ functions). Shilov might be a good name to search if you're looking for results like this (although you might know that already).

Ah - you already know about Grobner bases. I took too long to write that last comment!

I suspect you're probably looking for a resultant type thing to tell you which $P$s give you systems which are truly overdetermined - not really my area of expertise I'm afraid but I'll still help if I can :)

Oh I see what you mean (i.e. you're hoping that 'most' $P$s give you overdetermined systems with no solutions). However, the fact that you have more equations than unknowns doesn't necessarily make a system overdetermined - you'd also need the equations to be suitably 'independent' (so that each equation is genuinely a new constraint). I'm can't put my finger on quite what kind of 'independence' you'd need without thinking about it a bit more, but I think it's a promising approach.