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Results tagged with ap.analysis-of-pdes
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user 7294
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
25
votes
Motivation for and history of pseudo-differential operators
Let me put it this way. Many natural objects in PDE theory are pseudodifferential operators. Just a few examples (besides, obviously, differential operators):
1) singular integral operators in the se …
- 9,007
11
votes
Sobolev spaces and geometry
No time to give a complete answer but just a hint to a possible direction. Sobolev spaces in $R^n$ arise as the largest possible spaces on which some functional ('energy') can be defined. So they are …
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9
votes
Does Physics need non-analytic smooth functions?
Let me add to the several excellent answers. From a physicist' point of view, a concept like 'the value of a field at a point' does not have much meaning, or at least it can certainly not be measured. …
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8
votes
Accepted
Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
If $u_j$ is such a basis, from $(\nabla u_j,\nabla u_k)=0$ for all $j\not=k$ it follows $(\Delta u_j,u_k)=0$ which means $\Delta u_j$ is a multiple of $u_j$, and unfortunately $\Delta$ has no eigenval …
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8
votes
Accepted
A good reference for the wave front set
There are many references at various levels of difficulty; it also depends on what aspects are you interested in. I cite out of memory, so beware of inaccuracies (which can be corrected according to y …
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8
votes
Accepted
When is a distribution having a finite support actually zero?
Let $E$ be the square $(0,1)^2$ in $R^2$, $D=\partial_x\partial_y$ and $u=1$. The support of $D(\chi_E u)$ is the set of corners of $E$.
- 9,007
8
votes
Accepted
$L^p-L^q$ boundedness of this simple singular oscillatory integral operator
Let me consider instead the operator
$$Sf(x):=\int e^{-i x y} \frac{f(y)}{|x-y|^{\alpha}}dy$$
which has the same properties as $T$ since $Tf(x)=Sf(-x)$. Writing
$$e^{-i x y}=e^{i|x-y|^{2}/2}e^{-i|x|^{ …
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8
votes
What is dispersive estimate?
The intutitive picture is the following: there is a certain amount of "mass", which at time $t=0$ you might imagine concentrated in a heap near a location $x=0$. This heap evolves with time according …
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7
votes
Solitary waves and their symmetries
Palais' fantastic summary tells most of what is necessary to know if you want to understand solitons (and I re-read that paper with pleasure). I thought I'd add just one more perspective since the mor …
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7
votes
Accepted
Can the solution of an elliptic operator with smooth coefficients have zeros of infinite order?
Take a look at this paper by Protter; there are certainly newer references (which you can easily find starting from this one), but basically the problem was already solved at the time.
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7
votes
History of fundamental solutions
Concerning question 2: Lars Garding credits G.Tedone, in a 1898 paper in the first volume of (the third series of) Annali di Matematica, for the general solution formula for the wave equation. Also Ha …
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7
votes
Fundamental solution for a parabolic PDE with constant coefficents
For an operator of the form $L=\partial_t-\sum A_{jk}\partial_j\partial_k$, the fundamental solution is computed in Section 3.3 of Volume I of Hormander's treatise (The Analysis of Linear PDOs). I thi …
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6
votes
Accepted
$L^\infty$ estimate on heat equation with a lower order term
The estimate as stated is clearly false since $u\to u_0$ as $t\to0$ hence $\|u\|_{L^\infty}\ge \|u_0\|_{L^\infty}$.
Anyway you can write the kernel explicitly and extract the information you need fro …
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6
votes
Inverse of partial differential operator as a smooth tame map
An addendum to Deanne Yang's answer. Actually, if a linear hyperbolic equation is degenerate, meaning that the characteristic roots are real but not distinct, then the inverse operator presents a loss …
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6
votes