Questions tagged [differential-operators]

Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.

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Positive definite matrix and Hörmander theory

Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$ Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set $$ \varphi_{\...
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Invariant differential operators on homogeneous spaces

Let $G$ be a reductive complex algebraic group and $H$ a reductive algebraic subgroup. Then there is a well-known isomorphism $$\text{Diff}^G(G/H)\cong \text{Dist}(G/H,eH)^H$$ where $\text{Diff}^G(G/...
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Question about differential operators in a completely non-integrable distribution

Say I have two integrable codistributions $$ U = \langle du^1, \ldots, du^m \rangle, \qquad Z = \langle dz^1, \ldots, dz^N \rangle $$ on a manifold $M$, with $N >> m$. Suppose that the ...
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Second order differential operator with a Lipschitz coefficient

Let $a(x) \in W^{1, \infty}(\mathbb{R})$ be real-valued such that $a(x) \ge a_0 > 0$. Let $A^2$ denote the second order differential operator $A^2 : = -\partial_x (a(x) \partial_x) + 1 : L^2(\...
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On the convergence of operators and their spectra

We consider a sequence of operators $\{L_n\}_{n=1}^\infty$. Each operator $L_n$ is a densely defined (possibly unbounded) closed linear operator on a real Hilbert space $H_n.$ The domain of $L_n$ is ...
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Why are we interested in spectral gaps for Laplacian operators

Let $M$ be a Riemannian manifold and let $\Delta$ be its Laplacian operator. There is a large literature on a spectral gap for such a $\Delta$, that is, finding an interval $(0,c)$ which does not ...
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2 answers
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Harmonic polynomials on the sphere

Let $\mathbb{S}=\{x\in\mathbb{R}^n|x_1^2+\ldots +x_n^2=1\}$ be the unit sphere in $\mathbb{R}^n$, $\mathbb{C}[x]=\mathbb{C}[x_1,\ldots ,x_n]$ the complex-valued polynomial functions on $\mathbb{R}^n$, ...
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Application of the Frechet derivative [closed]

$f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differentiable at $x_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that: \...
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2 votes
2 answers
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Sharp Dirichlet heat kernel estimates in exterior domains?

I am right now working on some linear parabolic problems studying the behaviour of its solutions for large initial data. To do this, I have needed to use some estimates of the Dirichlet and Neumann ...
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Unbounded operator with closed range

Consider the spaces $C_{2\pi}^m$ of smooth $2\pi$-periodic functions, and the unbounded operator $L:C_{2\pi}^m\to C_{2\pi}$, given by $L(u)(t)=P(\partial)u(t)+u(t-\tau)$ where $P(\partial)$ is a ...
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Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$

Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
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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 ...
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2 votes
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Confusion about Wakimoto's chiral differential operators on $\mathbf{P}^1$

It is a classical result of Wakimoto that the sheaf of chiral differential operators $D_{ch}$ on $\mathbf{P}^1$ has global sections $$D_{ch}(\mathbf{P}^1)\ \simeq\ L_{-2}(\mathfrak{sl}_2)$$ the simple ...
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Was an index theorem for manifold with local boundary condition proven?

I would like to ask a question on the bibliography of the index theorems on manifold with boundary. Before my bibliographical research my understanding of the field was that for manifold with boundary,...
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Monotonicity of the top eigenfunction of the generator of a diffusion

Consider in 1D the operator given by $$ \mathcal{L} = \frac{d^2}{dx^2} - V'(x)\frac{d}{dx}, $$ where $V(x)$ is a convex, sufficiently quickly growing potential, so that $\mathcal{L}$ has a complete ...
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Time regularity of traces

I have a question about the time regularity of the traces in one dimension. Suppose I have a function space $$X = C^1([0,T];L^2(0,1))\cap C([0,T],H^1(0,1))$$ and I define an operator $E$ on $X$ by $(...
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On which subspace $W\subset C^{\infty}[0,1]$ is $(Df)(x)=xf'(x)$ a bounded operator provided all functions in $W$ are flat functions?

I have already asked this question on MSE; now I repeat it on MO. https://math.stackexchange.com/questions/4132346/on-which-subspace-w-subset-c-infty0-1-is-df-xfx-a-bounded-operator First we ...
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4 votes
1 answer
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Domain of Friedrichs extension of $-\partial^2_r + mr^{-2} : L^2(0,\infty) \to L^2(0,\infty)$

Consider the second order differential operator $$ A = -\partial^2_r + mr^{-2} : L^2(0, \infty) \to L^2(0,\infty), \qquad m \ge -\frac{1}{4}, $$ equipped with domain $C^\infty_0(0, \infty)$. Since $\|...
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3 votes
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Reference request for spectral theory of elliptic operators [closed]

I want to learn the spectral theory of linear elliptic operators in bounded and unbounded domains in $R^n$, in particular for Laplacian and Schrodinger operators. Please suggest me some reference. I ...
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On a harmonic coordinate for a differential operator

I am looking for a harmonic coordinate for a degenerate differential operator on the unit ball. We write $\langle \cdot,\cdot \rangle$ for the standard inner product on $\mathbb{R}^d$. Set $\lvert\...
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Taylor series with less than differentiability

I have a function $f^0\colon (0;\infty) \to \mathbb R$ with the property that the following limit exists and is finite $$ F^1 := \lim_{x\to \infty} x \cdot f^0(x) $$ Then I consider $f^1(x) := x \cdot ...
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Is it possible in principle (but not in practice) to recursively factor away the Riemann zeta zeros as they are computed?

Let: $$f_0(x)=\frac{\zeta (x)}{\sin \left(\frac{\pi x}{2}\right)}$$ and let the seed point be: $$s=\sqrt{-1}$$ which is the input into the limit: $$\rho_1=s+\lim\limits_{n \rightarrow \infty}\left(1-\...
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A question about Dirac operators

Let $D$ be a Dirac operator on spinor bundle $S$ over even-dimensional non-compact spin manifold $X$, $$ \left<s_1,s_2\right>_{L_2} = \int_X \left<s_1,s_2\right> \quad \forall s_1,s_2\in\...
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2 votes
1 answer
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Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"

(This question is related to Splitting a space into positive and negative parts but different.) Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...
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Regularity of $\frac{d}{dt}f = Df + Bf$ in the interior of a cylinder

Suppose that $f\in L^2_1([0,1]\times \mathbb{T}^m)$ satisfies the following PDE $$\frac{d}{dt}f = Df + Bf$$ $$f(0) = g\in L^2_{3/2}(\mathbb{T}^k)$$ where $D:L^2_1(\mathbb{T}^m)\to L_0^2(\mathbb{T}^m)...
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Relationship between point spectrum and generalized inverses(Green operators) of elliptic self-adjoint operators

I'm very confused about the relationship between point spectrum and generalized inverses(Green operators) of elliptic self-adjoint operators. I'm following proof of a theorem in a paper, it seems that ...
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3 votes
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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 ...
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A question about parametrix

$\DeclareMathOperator\id{id}$Let $D$ be a differential operator, and $Q$ a parametrix of $D$, i.e., $$ QD=\id-S_0 ,\qquad DQ=\id-S_1$$ where $S_0$, $S_1$ are operators with smooth kernels. A special ...
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Closure of $f\mapsto\sigma f''$ on $\mathcal{C}^2(\,[0,1]\,)$

Let $\sigma\in\mathcal{C}^0(]0,1])$ a positive function such that $\lim\limits_{t\rightarrow 0}\sigma(t)=0$, and $f\in\mathcal{C}^0(\,[0,1]\,)\cap\mathcal{C}^2(\,]0,1]\,)$ such that $\lim\limits_{t\...
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Non-linear, hyperbolic, 2nd order system of PDEs

This is a cross-post. In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system \begin{...
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1 vote
1 answer
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Derivatives and exponential derivatives quotient operators on two variables

I consider for example the following function of two variables given by $$f(x,y)=\sum_{n=0}^{+\infty}\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}\left(\frac{x}{y}\right)^{n}...
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5 votes
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Center of Grothendieck differential operators in positive characteristic

Let $k$ be a field of characteristic $p$. Consider the algebra $A:=\mathcal{D}(k[x])^{S_2}$ consisting of Grothendieck differential operators invariant under the $S_2$ action $x\mapsto -x$. The ...
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3 votes
1 answer
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Twisted differential operator, chiral differential operator, $???$ (continue the sequence)

Let $X$ be a smooth variety. One can define the notion of a sheaf of twisted differential operators (TDO) on $X$. They "quantise" functions on $T^*X$. Examples include the usual sheaf of ...
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A problem involving a highly non linear system of PDE

Is there a unique smooth function $u:\mathbb{R}^3:\rightarrow \mathbb{R}^3$ such that $$ \begin{eqnarray} \dfrac{\partial}{\partial t}\left(u_1^2+u_2^2+u_3^2\right) & = & \mu \left(u_1\dfrac{\...
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Function annihilated by ideal in universal envelope

Let $G$ be a Lie group, $U(G)$ its universal enveloping algebra over $\mathbb C$ and let $J\ne U(G)$ be a left ideal. We consider $U(G)$ as the algebra of left-invariant differential operators on $G$. ...
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1 vote
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Hypoellipticity of a heat-like parabolic operator on Riemannian manifolds - reference request

Let $(M,g)$ be a Riemannian manifold and $L$ be a differential operator on $M$, with smooth coefficients, such that its symbol be $g$ (a "generalized Laplacian"). Where can I find proved ...
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5 votes
1 answer
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Is the second weak derivative a self-adjoint operator?

Let $W^{2,2}(\mathbb R)\newcommand{\R}{\mathbb R}\newcommand{\C}{\mathbb C}\newcommand{\N}{\mathbb N}$ denote the Sobolev space as defined in chapter 5 of Evans' PDE book and consider the linear ...
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Propagation of Klein-Gordon solutions in extra dimensions

In his paper "Von Neumann Algebras of Local Observables for Free Scalar Field" Araki used the solutions of the equation $$\frac{\partial ^{2}h}{\partial x^2}-\frac{\partial ^{2}h}{\partial t^...
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3 votes
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The exponential derivative operator

Thank you very much for the interesting responses in my previous question The Quotient exponential operator. I have another complicated formula related to the previous one in the following $$ \exp\...
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7 votes
2 answers
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Differentiation of functions over graphs

In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary. Definitions. Let $G=(V,E)$ be a directed ...
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3 votes
1 answer
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Differential inequalities under which a flat function must be identically zero

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
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5 votes
2 answers
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Line graphs called "graph derivatives": any intuition?

Short version: in several papers, line graphs (and closely related graphs) are called graph derivatives or derived graphs; is there any intuition for such terminologies, in connection with the ...
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3 votes
0 answers
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Another question from Villani's monograph "Hypocoercivity"

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 62 (the first snapshot), he defined a new inner product $((\cdot,\cdot))$ as in (...
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A question on Villani's monograph "Hypocoercivity"

I can not figure out the appearance of the term $\int h_0\,d\mu$ in the statement of Theorem 35 above. Here are some background information: $L$ is an unbunded operator on a Hilbert space $\mathcal{H}^...
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A possible error in Villani's monograph "Hypocoercivity"

I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 48, he wrote "For the second term in (6.9), we use the identity $$\nabla\...
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  • 410
5 votes
1 answer
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Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...
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3 votes
1 answer
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Ricci flow for manifold learning

I know that mean curvature and diffusion-type flows are common in manifold learning because of their smoothing effects. I haven't seen Ricci flow used as much. Given that Ricci and diffusion-type ...
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Range of divergence operator on the space of traceless symmetric $(0,2)$ tensors; conformal vector fields on an arbitrary metric on $S^2$

Let $\gamma$ be a metric on $S^2$. I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$: $$div_{\gamma} A = \omega$$ where $\omega$ is a 1-form. It is known that there ...
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3 votes
1 answer
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Is it possible that $\int |f^2(z)|^{t+1}(P\bar{P})\phi(z) dz=0$ for all compactly supported $\phi$?

When proving that the log-canonical threshold is minus the largest root of the Bernstein-Sato polynomial one considers the integral $$\int |f^2(z)|^{t+1}(P\bar{P}(\phi(z))) dz,$$ where $P$ is a linear ...
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  • 2,655
4 votes
1 answer
163 views

Degenerate second-order Lagrangians

Let $M$ be a smooth $m$ dimensional manifold, let $\pi:E\rightarrow M$ be a smooth fibred manifold over $M$. Let us write generic fibred coordinates as $(x^i,y^\sigma)$ with $x^i$ being the base ...
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