Questions tagged [pseudo-differential-operators]
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.
166 questions
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Calculi of pseudodifferential operators and K-theory
I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...
- 319
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49
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On the $L^p$ estimate and Weyl's law of Eigenfunctions in Sogge's Book
I have recently started to study the book "Fourier integrals in classical analysis " by Sogge mainly oscillatory integral decay methods. I have a question from the chapters 4 and 5. Mainly ...
- 11
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1
answer
121
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Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta<\infty$, but $u\notin H^{1/2}$
For $u\in \mathcal{S}'(\mathbb{R})$, define, if finite,
$$\Lambda(u)^4=\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta.$$
Using the triangle inequality $|\eta-\xi|\le |\eta|...
- 850
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162
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Dual space of local Sobolev space on a manifold
$\newcommand{\comp}{\mathrm{comp}}$As part of my master's thesis, I am currently learning about Sobolev spaces on manifolds. From my research online, I found out, that there are a lot of ways to ...
3
votes
1
answer
128
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Fréchet-valued symbols
Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$...
- 395
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70
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About Fourier integral operators
Consider the operator
$$(A_\Delta\psi)(p) = \int_\Delta \int_{\mathbb{R}^n} e^{ix\cdot(p-k) + i\phi(x,p,k)} a(x,p,k) \psi(k) dk\: dx$$
where $\Delta$ is a Borel or Lebegue set in $\mathbb{R}^n$, in ...
- 205
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82
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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
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1
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133
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$L^p$ boundedness for pseudo-differential operators
Let $\rho, \delta, m$ be real parameters such that $0\le \delta\le \rho\le 1, \delta<1$. The set $S^m_{\rho, \delta}(\mathbb R^{2n})$ is defined as the set of smooth functions $a$ on $\mathbb R^n\...
- 16.2k
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247
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Leibniz rule for square root of Laplacian
Let $(M,g)$ be a compact Riemannian manifold (e.g. $M=S^3$ the 3-sphere) and let $\Delta$ be the metric Laplacian on $M$. Then $\Delta$ has an $L^2(M)$ basis of eigenfunctions $\pi_m$, $$ \Delta \pi_m ...
- 914
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207
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Differential operators in $\Bbb R^n$
Put $P_j=\frac{\partial}{\partial \xi_j}$ et $Q_j=2 i \xi_j$ with$\xi=\left(\xi_1, \ldots, \xi_n\right)$ et $x=\left(x_1, \ldots, x_n\right)$. How to prove :
$\exp \left(\sum_{j=1}^n x_j P_j\right)(...
- 511
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vote
1
answer
183
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Closed form of a Fourier transform
I apologize for not being able to motivate the question below; it would go into technicalities.
Let $n=d+1\ge2$ be the space-time dimension, and
$$H(y,t):=\left(\frac{t^2}{(t^2+|y|^2)^{1+d/2}}\right)^{...
- 52.3k
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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
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Differential form of the multidimensional "orthogonal dilation" operator
For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion.
...
- 286
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355
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Real-analytic analogue of Schwartz functions
Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic ...
- 53
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80
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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
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304
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Is it possible to define pseudodifferential operator $p(x,T)$ using Cauchy integral formula?
I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.
Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:
$$\...
- 350
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106
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Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform
Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
- 350
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104
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Kernel representation of a power of (pseudo-)differential operator
Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation:
\begin{equation}
\mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt.
\end{equation}
What can ...
- 350
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309
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Bounded operator on $L^2(\Bbb R^2)$
Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$:
$$
f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...
- 511
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1
answer
311
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Conormal distributions and the wave front set
Let $X$ be a smooth closed manifold and $Y$ a regular submanifold. For all conormal distributions at $Y$ on $X$, their wave front set is contained in the conormal bundle of $Y$. Is the reciprocal true?...
2
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85
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Is there an easy explanation for the notion of Calderon projectors?
I recently stumbled upon the notion of Calderon projectors, which are standard tools for elliptic boundary value problems. However, when searching on the internet I could not really find any useful ...
- 1,171
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1
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87
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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
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0
answers
129
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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
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2
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352
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Reference for commutator estimate
I'm interested in Sobolev space estimates for commutators involving a pseudodifferential operator and a Fourier multiplier. More specifically, suppose $p = p(x,\xi) \in S_{1,0}^{m_1}$ and let $q = q(\...
- 683
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320
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Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
- 395
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1
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131
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Positivity of an operator on a compact subset of a manifold
Let $E$ and $F$ be two vector bundles over manifold $X$. Let $P:\Gamma(E)\to \Gamma(F)$ be a self-adjoint differential operator over $X$. Define inner product on the spaces $\Gamma(E)$ of smooth ...
- 563
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151
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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_{\...
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67
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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
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244
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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 $C^{(1, \frac{1}{3})}$ ...
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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, ...
4
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310
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Sobolev spaces and spectral theorem
Consider a generalised harmonic oscillator of the form
$$
A(x,D_x) = \langle x \rangle (1-\Delta_x)\langle x \rangle, \quad x \in \mathbb{R}^n,
$$
where $\langle x \rangle = (1+|x|^2)^{1/2}.$ The ...
1
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0
answers
67
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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
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1
answer
195
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Metric / strong slope restriction of function on unit ball in $\mathbb R^m$
Diclaimer. I'm not sure this is the right venue for this question, but I'll give it a try
Definition [Strong / metric slope]. Given a complete metric space $(M,d)$ and a function $f:M \to (-\infty,+\...
- 6,853
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98
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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
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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 ...
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0
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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
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171
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Pseudodifferential Operators and Functional Calculus
I hope this is not too naive a question for MO. I've been taking a mathematical physics course, and was shown how operators like $\sqrt{1-\Delta}$ could be defined by taking multiplication operators ...
- 737
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0
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312
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History of microlocal analysis
Was the study of pseudo-differential operators the basis for the birth of microlocal analysis? I'm trying to find out how this branch of analysis was developed...
- 97
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197
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Pseudo-differential operators and differential operator
I am totally new to pseudo-differential operators and I’m wondering if a differential operator is a pseudo-differential operator.
So, I want to show , using the definition of the symbol given by ...
- 97
1
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1
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157
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Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$
How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?
- 69
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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
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1
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193
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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
$$\partial_t f +v\cdot\nabla_x f \pm c\nabla(-\Delta)^{-1}...
- 1,247
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3
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644
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Question about the regularity of fractional Heat equation
Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation:
$$
\...
- 371
3
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0
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91
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Pseudodifferential operator associated to a self-adjoint extension of a symmetric operator on an incomplete manifold
Let $D$ be the Dirac operator acting on a spinor bundle $S$ over a complete Riemannian manifold $M$. Then $D$ is an essentially self-adjoint operator on $L^2(S)$.
Suppose there is a compact subset $K\...
- 1,903
1
vote
1
answer
852
views
Kernel of the composition of operators
Let $X \subset \mathbb{R}^{n}$, $Y \subset \mathbb{R}^{m}$, and $Z \subset \mathbb{R}^{p}$ be open subsets, and let $K_P \in C_0^\infty(X \times Y)$ and $K_Q \in C_0^{\infty}(Y \times Z)$. Then, $K_P$ ...
- 509
1
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1
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153
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Extension of pseudodifferential operators to Sobolev spaces
Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and define \begin{eqnarray*}
\mathcal{S}^m(\Omega ) &=&\left\{
\begin{array}{c}
a\in C^\infty(\Omega \times \mathbb{R}^{d}): \hbox{for all ...
- 509
1
vote
0
answers
55
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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
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1
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260
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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
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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
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1
answer
1k
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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