All Questions
Tagged with pseudo-differential-operators ap.analysis-of-pdes
62 questions
3
votes
0
answers
82
views
Dirichlet-to-Neumann map is analytic
Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
- 2,095
1
vote
0
answers
97
views
Oscillatory integrals and regularity
Let $U\subset\mathbb{R}^{d}$ be open and $N\in\mathbb{N}$. Furthermore, let $a\in\mathcal{S}^{m}_{\rho,\sigma}(U\times\mathbb{R}^{N})$ be a symbol and $\Phi\in C^{\infty}(U\times (\mathbb{R}^{N}\...
- 1,171
7
votes
0
answers
80
views
Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator
Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
- 808
-1
votes
1
answer
87
views
Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$
I am having the following integral:
$$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$
where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$,
$...
- 159
3
votes
0
answers
129
views
Reasons behind different conventions for symbol of operator
I've come across two slightly different conventions for the symbol of a differential operator $D$ (let's say on $\mathbb{R}^n$) and haven't thought much about the motivation behind them until now.
The ...
- 1,903
3
votes
1
answer
151
views
Symbol estimates using metric on the phase space
Consider the symbols introduced by Beals and Fefferman in which the symbol estimates are of the form:
\begin{equation}\tag{1}
\label{eq1}
|\partial_x^\beta\partial_\xi^\alpha a(x,\xi)| \leq C_{\...
2
votes
0
answers
67
views
Asking a reference for a fact about nonlocal operators
Let assume that $u$ is smooth enough and $ -\Delta (u \phi) \in L^1(\Omega)$ for any $\phi \in C_c^{\infty}(\Omega)$. Then it easily follows that $ -\Delta u \in L^1_{\mathrm{loc}}(\Omega)$ by ...
- 371
0
votes
0
answers
244
views
Holder continuity relative to Rellic-Kondrachov compactness via the nonlinear Aronsson operator
Connected to the question,
Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?
An analysis of the well-known nonlinear Aronsson operator gives $C^{(1, \frac{1}{3})}$ ...
1
vote
0
answers
323
views
Does Morrey's inequality contextually relate to Rellic-Kondrachov compactness?
I have been reflecting on this question, and want to share my thinking thus far. I'd be grateful for the community's inputs.
We refer to Morrey's inequality, Theorem 4 on pp 266 of Evan's book on PDE, ...
1
vote
0
answers
67
views
Regularity and existence linear parabolic fractional equation
\begin{equation}
\begin{cases}
a(x, t)u_t+(-\Delta)^{\sigma}u+b(x,t)u=f(x,t), & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x), & \text{in } \mathbb{R}^n
\end{cases}
\end{...
- 11
3
votes
0
answers
98
views
Semiclassical analysis and reflection law
I was curious about the following vague question. In pseudodifferential calculus and semi-classical analysis there are various theorems that relate geodesics of the underlying space to the behavior of ...
- 445
5
votes
1
answer
2k
views
Does the 'reproducing kernel formula' for a bounded open set $U$ define an equivalent norm on the Sobolev space $H^1_0(U)$
We refer to the 'reproducing convolution formula with a kernel' for an open bounded domain $U$ of $R^n$, $n \geq 2$ discussed in the paper of G. Talenti (Annali de Matematica, Dec 1976) on Best ...
1
vote
0
answers
74
views
Well-posedness for a wave equation with degenerate coefficients
Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem:
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
t\partial_t(t\partial_t u)-\...
- 4,135
2
votes
0
answers
159
views
Principal symbol of a non-local operator and Atiyah–Singer index formula
I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
- 41
1
vote
1
answer
193
views
Reference for singular integral operators such as $(-\Delta)^{-1}$ or $\nabla(-\Delta)^{-1}$
I'm currently working on understanding certain mean-field limits in kinetic theory, and the equations I'm working with are usually of the form
$$\partial_t f +v\cdot\nabla_x f \pm c\nabla(-\Delta)^{-1}...
- 1,247
1
vote
0
answers
55
views
The order of regularity improving for elliptic operator with rough coefficients
Let $\frac12<\alpha<1$ and let $L(x)$ be a first order overdetermined elliptic operator with coefficients in $C^\alpha$. We means $L(x)=\sum_{j=1}^nA_j(x)\partial_j$ where $A_j:\mathbb R^n\to\...
- 938
2
votes
1
answer
260
views
Metric on the phase space
I am studying PDEs whose symbols satisfy
\begin{equation}
|\partial^\alpha_\xi\partial^\beta_xp(x,\xi)| \lesssim M(x,\xi)\Psi(x,\xi)^{-|\alpha|}\Phi(x,\xi)^{-|\beta|}
\end{equation}
for all multi-...
2
votes
1
answer
473
views
Question on definition of Dirichlet to Neumann operator
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with a
$C^2-$ boundary $\partial \Omega= \Gamma$. For $f \in
H^{1/2}(\Gamma)$, let $F \in H^1(\Omega)$ denote the weak solution of
the ...
- 227
11
votes
1
answer
1k
views
Research topics in microlocal analysis
Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
- 589
1
vote
0
answers
76
views
PDE on an open ball with prescribed value on some open subsets
Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
- 2,533
2
votes
0
answers
315
views
Support of a microlocal defect measure
I'm trying to complete the proof of the Theorem 6.1 in the notes https://www.math.u-psud.fr/~nb/articles/coursX.pdf, which ensures, under certain conditions, that the support of the microlocal defect ...
- 509
3
votes
0
answers
114
views
Parametrix of external product of elliptic operators
Let $S$ and $T$ be Dirac-type operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. ...
- 1,903
2
votes
1
answer
127
views
Positive form for a homogeneous elliptic pde
I have a pde of the following form:
\begin{align}
&P(x,D)u = f \text{ on } \Omega, \\
&P(x,D) = \sum\limits_{|\alpha|=2m}a_\alpha(x)D^{\alpha},
\end{align}
where one can assume that $f$ ...
- 537
0
votes
1
answer
175
views
Accessible reference for (scattering) $\Psi DO$'s on manifolds
I am currently trying to understand Hassell, Tao, and Wunsch's paper on Strichartz estimates on non-trapping asymptotically conic manifolds, however, my understanding of pseudodifferential operators ...
- 1,247
3
votes
0
answers
588
views
Propagation of Singularities
I'm following the book "Elementary Introduction To The Theory Of Pseudodifferential Operators" by X. S. Raymond and the Joshi Lectures Notes - https://arxiv.org/pdf/math/9906155.pdf - to prove the ...
- 509
2
votes
0
answers
40
views
Production of $H^s$ singularities in the strictly hyperbolic Cauchy problem
This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without ...
- 503
7
votes
2
answers
517
views
Do pseudo-differential operators form a sheaf of algebras?
Let $M$ be a smooth manifold.
I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on ...
- 7,789
2
votes
3
answers
541
views
BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?
I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
- 159
14
votes
4
answers
1k
views
Is every continuous microlocal operator a pseudo-differential operator?
Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space.
Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal.
By being microlocal I mean that the wave ...
- 8,086
11
votes
1
answer
668
views
Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?
Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
- 7,789
5
votes
1
answer
301
views
$L^p$-estimates for elliptic pseudodifferential operators
Assume we have an pseudodifferential operator
$P:\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S}(\mathbb{R}^n), Pf(x) = (2\pi)^{-n/2}\int\mathrm{d}\xi\; p(x,\xi)\,\hat{f}(\xi)e^{i\xi x}$
acting on ...
- 381
0
votes
1
answer
104
views
Poisson Equation across a Hypersurface [closed]
Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem
$ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
- 4,135
2
votes
1
answer
244
views
Smoothness of distributions defined by oscillation integrals
In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement.
Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...
- 1,247
1
vote
0
answers
180
views
Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
- 115
2
votes
0
answers
240
views
Proof of Taylor's Schwartz kernel estimate of pseudodifferential operators
I am interested in the proof of the following result which gives an estimate on the Schwartz Kernel of a $\Psi$DO. There is one aspect the proof that is not clear to me which I would like to ask the ...
- 53
1
vote
0
answers
61
views
Convergence of PDE/PsiDE - expansion of pseudo-differential operators
I have am working with a nonlinear pseudo-differential evolution equation of the form
$$u_t + \mathcal{N}(u) + \mathcal{D}_{\epsilon} u = 0$$
where $\mathcal{N}$ is a nonlinear operator and $\mathcal{...
- 155
1
vote
0
answers
70
views
Normal form of Principal type $\Psi$DO's
Suppose we have a pseudo differential operator of principal type with a complex symbol and such that the poisson bracket of the real and imaginary parts on the characteristic set is non-negative.I ...
- 4,135
4
votes
0
answers
2k
views
Fourier transform of $C^\infty_0$, smooth functions vanishing at infinity
Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \...
- 41
4
votes
0
answers
237
views
Contact manifolds and pseudodifferential operators
By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...
- 41
1
vote
1
answer
378
views
Easy Garding Inequality
Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...
- 4,135
4
votes
1
answer
161
views
Commutator representation of certain smoothing operators
I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...
- 313
5
votes
1
answer
270
views
Positivity of semiclassical pseudodifferential operators
Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis)
Let $m\in\mathbb{R}$, and $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\...
- 305
2
votes
0
answers
188
views
Hypoelliptic pseudodifferential operators and Fredholm equations?
I read here
the following:
The parametrix is a useful concept in the study of elliptic
differential operators and, more generally, of hypoelliptic
pseudodifferential operators with variable ...
- 383
9
votes
2
answers
620
views
Pseudodifferential operators on spaces with boundary
Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann ...
- 93
1
vote
0
answers
77
views
Recursive formula for symbol of resolvent on noncompact manifold
On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: ...
- 313
3
votes
1
answer
811
views
Application of Egorov's Theorem for Pseudodifferential Operators
Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic ...
- 55
12
votes
1
answer
1k
views
applications of C$^*$-algebras in the field of PDEs
I know only a little bit about C$^*$-algebras and I want a to know if you know a nice apllication or the influence of them in the field of partial differential equations (it is better that it is ...
8
votes
0
answers
278
views
Pseudodifferential operators on compact manifolds with boundary
I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
- 81
3
votes
1
answer
398
views
Elliptic Fourier integral operators
I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point.
But ...
- 31
8
votes
1
answer
712
views
Pseudo-differential operators with compactly supported symbols
If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO $...
- 728