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Results tagged with differential-operators
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user 121820
Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.
5
votes
Accepted
Hodge decomposition for non-elliptic complexes
This is false. Equip $T^2 = S^1 \times S^1$ with the Lorentzian metric $g = -ds^2 + dt^2$. For concreteness, regard $S^1 = \mathbb{R} / \mathbb{Z}$. Consider the de Rham complex. The Hodge Laplaci …
- 1,713
6
votes
Simplification of integral on the sphere
You can also derive the lemma from the Bochner formula, which in general dimensions can be written
$$ \frac{1}{2}\Delta\lvert\nabla u\rvert^2 = \lvert\nabla^2u\rvert^2 - (\Delta u)^2 + \delta\left((\ …
- 1,713