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Results tagged with pseudo-differential-operators
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user 7333
This tag is for questions regarding to the Pseudo-differential operator, acting on a space of functions on a differentiable manifold, that can locally be described by definite rules using a certain function. It satisfies estimates for the derivatives analogous to the estimates for derivatives of polynomials, which are symbols of differential operators.
3
votes
Functions of pseudodifferential operators
Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:
As pointed out by Liviu Nicolaescu in th …
- 2,821
9
votes
Motivation for and history of pseudo-differential operators
A slightly different motivation for fourier integral operators and pseudo-differential operators is given in the first chapter of this book - Fourier Integral Operators, chapter V. W. Guillemin: 25 Ye …
- 4,713
11
votes
2
answers
1k
views
Why take 'complex powers' of pseudo-differential operators?
Given a pseudo-differential operator $P$ of order zero, Seeley showed that the holomorphic family of operators $\lbrace P^{z} : z\in \mathbb{C} \rbrace$ of all complex powers is contained in the clas …
- 308
4
votes
Why is symplectic geometry so important in modern PDE ?
As the cotangent space has a natural symplectic structure, we can also phrase the question as "Why is cotangent space so important in modern pde?". A nice answer for this question is given in http://m …
- 2,239
7
votes
1
answer
867
views
Pseudo-differential operators which are independent of lower order perturbations
In the area of pseudo-differential operators, we know that for elliptic type or real principal type operators reductions are independent of lower order terms. For example, if $P$ is zeroth order real …
CommunityBot
- 1
3
votes
2
answers
214
views
a question about first-order hyperbolic equations
Performing certain manipulations on pseudo-differential equations I have come across the following first order equation:
$$
D_{t}u-\lambda(t,x,D_{t},D_{x})u=0, \ \ (*)
$$
where $\lambda$ is a scalar p …
- 27.5k
1
vote
second order operator with real coefficients and not locally solvable
The following example is an second order differential operator with real coefficients defined on $\mathbb{R}^{3}$ which is not solvable about origin:
$Pu=(x_{2}^{2}-x_{3}^{2})D_{1}^{2}u+(1+x_{1}^{2}) …
- 2,239
4
votes
2
answers
247
views
second order operator with real coefficients and not locally solvable
This question is a follow up of the following answer I posted recently:
Counterexamples in PDE
Is there a second order partial differential operator with real coefficients which are not solvable in …
CommunityBot
- 1
7
votes
Why is symplectic geometry so important in modern PDE ?
Fourier Integral Operator is an operator which has its Schwartz kernel as a distribution whose singularities are on a Lagrangian submanifold. In fact, we can associate a FIO with an amplitude and a La …
- 2,239