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bio website math.poly.edu/~yang
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visits member for 4 years, 11 months
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Sep
21
comment Differentiability of Nemytskii operator on Sobolev space
It seems like you're on the right track. It's just the chain rule, no?
Sep
21
comment Did differential geometry undergo a notation change?
Thomas, that's a really good point! The ordering of the indices of the curvature tensor is a confusing issue. It took me a long time to figure out the "right" way to order the indices and how to justify it.
Sep
21
revised Did differential geometry undergo a notation change?
added 4 characters in body
Sep
21
comment Estimate of a Sobolev norm of p-form
Your estimates are not stated correctly. I also suggest that you migrate the question to math.stackexchange.com .
Sep
21
revised Did differential geometry undergo a notation change?
added 492 characters in body
Sep
21
comment Did differential geometry undergo a notation change?
And if you think this is confusing, Kahler geometry is way, way worse, because you have to keep track not only of signs but also powers of $i$ and $2\pi$. Sometimes, the exact constant is not important but it's kind of upsetting if you're used to getting exact formulas. And sometimes it really matters.
Sep
21
comment Did differential geometry undergo a notation change?
My recollection is that there are only two major differences: There's the difference in sign, and he wrote all indices as subscripts when there was a metric involved. Otherwise, it looks the same, so you just learn roughly how the calculation goes and then do it yourself using your own notation.
Sep
21
answered Did differential geometry undergo a notation change?
Sep
17
revised Inverse of partial differential operator as a smooth tame map
edited title
Sep
17
answered Inverse of partial differential operator as a smooth tame map
Sep
17
comment Local solvability of nonlinear elliptic boundary value problems
After that, the theorem follows by the standard implicit function theorem for Banach spaces. The boundedness of the left inverse should follow by standard a priori estimates, using Sobolev norms, for elliptic boundary value problems. If for some reason the left inverse is not bounded but maps back into a space of functions with lower regularity than those in $E$ (i.e., it loses derivatives), then you can always use the Nash-Moser implicit function theorem. But it seems unlikely to me that this is needed.
Sep
17
comment Local solvability of nonlinear elliptic boundary value problems
Have you tried just adapting Malgrange's original proof? The idea is to identify two Banach spaces $E$ and $F$ of functions on the closure of U such that the linearized operator $P$ is a surjective bounded linear operator from $E$ to $F$ and there exists a left inverse of $P$ that's a bounded linear operator from $F$ back to $E$. To achieve surjectivity of $P$, you might have to shrink the domain sufficiently.
Sep
16
comment Normal-like coordinates for weakly differentiable metrics
Sorry. What you want to do is make the angular dependence singular b
Sep
13
comment Normal-like coordinates for weakly differentiable metrics
I suggest trying to see if this is true for simple examples, before trying to prove the general case. For example, $$g_{ij} = (1+|x|^{2-\alpha})\delta_{ij}$$ where $\alpha > -n/p$.
Sep
11
comment Poincare lemma for non-smooth differentiable forms
Another thing to try are harmonic forms, but this fails, too.
Sep
11
comment Poincare lemma for non-smooth differentiable forms
I agree with Jochen. In particular, I see how to use the standard proof of the Poincaré lemma to get exactness if the same $k$ is used for every term in the sequence. But I don't see how to recover another order of differentiation when proving surjectivity of the last map in the 2d exact sequence given in the question.
Sep
6
answered How is the metric tensor related to the Hessian of the first fundamental form?
Sep
5
comment How is the metric tensor related to the Hessian of the first fundamental form?
What do you mean by the "first fundamental form? Or its Hessian? As far as I know, the term "first fundamental form" is used mainly for a submanifold of Euclidean space and refers to the induced Riemannian metric on the submanifold.
Sep
3
awarded  ap.analysis-of-pdes
Aug
11
comment Writing papers in pre-LaTeX era?
Sorry but no way. And it might have been Swiss.