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Feb
3
comment Asymptotic behavior of a function
What is $\omega_1$? (In particular, what values does it take? When $n = 2$ is $\omega_1 \in [-1,1]$ or is it in $[0,2\pi)$?)
Feb
3
awarded  Good Answer
Feb
3
answered Applications of Gauss-Bonnet theorem
Feb
3
comment Linearized stream function
some arbitrary function $f$ [e.g. the case of traveling solitons]. But if you expect wave like behaviour than it is natural to postulate that $f$ takes the form of $\exp i \alpha$ and see what happens.)
Feb
3
comment Linearized stream function
@charlestoncrabb: "why it works" is basically answered by "you plug it in and see what happens". The motivation behind the choice is that parallel shear flow can be imagined to be a flow between two plates, where the fluid is flowing in the $x$ direction with velocity depending on the $y$ direction only. If you add a perturbation, you sort of intuitively expect the perturbation to be carried down stream in a traveling wave. So a first order approximation is precisely of the form $\Psi(y) \exp (i \alpha (x - ct))$. (A traveling wave solution more generally should depend on $f(x-ct)$ for
Feb
2
revised Boundary regularity of solution to partial differential equation
edited tags
Feb
1
comment The inverse of Laplacian operator for different orders
$I - \triangle$ is diagonal if you expand a function in terms of the eigenfunctions of the Laplacian. Functional calculus works basically the same way as that for self-adjoint operators on finite dimensional inner product spaces. If you like: secretly we are taking the "Fourier transform", except we are expanding using a basis adapted to the Laplacian on the domain.
Feb
1
comment The inverse of Laplacian operator for different orders
(@ChristianRemling: sorry to have wasted your time; I hid my answers while I was editing since the OP changed the question after I posted a computation along the lines of what you suggest.)
Feb
1
comment The inverse of Laplacian operator for different orders
@tankonetoone: for your first comment: use that eigenfunctions of the Laplacian with different eigenvalues are orthogonal. And just compute.
Feb
1
revised The inverse of Laplacian operator for different orders
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Feb
1
comment The inverse of Laplacian operator for different orders
For your final box: you can expand $u$ in terms of sum of eigenfunctions of Laplacian (with fixed, say Dirichlet, boundary conditions). The eigenfunctions are orthogonal to each other in $L^2$. The $\int u = \int v$ condition can be trivially satisfied by assuming $\int u = \int v = 0$ which is the same as saying that they have no $0$-frequency components.
Feb
1
revised The inverse of Laplacian operator for different orders
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Feb
1
answered The inverse of Laplacian operator for different orders
Feb
1
answered Linearized stream function
Jan
31
comment Marginally Trapped surfaces with constant Gaussian curvature
From a pure geometry point of view: it would be kinda neat to have a full classification of all round marginally trapped surfaces in de Sitter, especially since you claimed there are ones not generated from symmetry considerations.
Jan
29
answered Marginally Trapped surfaces with constant Gaussian curvature
Jan
25
awarded  Good Answer
Jan
21
revised PDE characterisation of elementary symmetric functions?
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Jan
21
revised PDE characterisation of elementary symmetric functions?
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Jan
21
answered PDE characterisation of elementary symmetric functions?