# Tagged Questions

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

Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap ... 1answer 150 views ### [This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale I have a stupid question: Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space$H^{-s}(\partial \Omega)$? We usually encounter trace theorem as ... 1answer 158 views ### derivative of the adiabatic limit of the eta invariant To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if ... 2answers 80 views ### Density of Laplace image Let$D$define the differential operator$-\frac{\partial^2}{\partial x^2}$on$\mathbb R$. Let$\xi\notin\mathbb R$be a complex number. Is it true that $$(D-\xi)C_c^\infty({\mathbb R})$$ is dense ... 1answer 143 views ### Ask the validity of Tauberian lemma in Sogge's book In C.D.Sogge's Fourier Integrals in Classical Analysis pp.128-129, he proved Lemma4.2.3(Tauberian Lemma): Lemma. Let$g(\lambda)$be a piece-wise continuous tempered function of$\mathbb{R}$. Assume ... 0answers 43 views ### what and where is Dubrovin's tri-differential operator? IFF memory serves, Dubrovin made good use of a tridiffereential operator long before H & R fluxes. What did he use it for and where? 1answer 73 views ### Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed] I have a real symmetric$3\times3$matrix$\mathbf{M}(\mathbf{r}$) which depends on$\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field$e_i(\mathbf{r})$over$\mathbb{R}^3$... 1answer 844 views ### Harmonic spinors on closed hyperbolic manifolds Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial? I'm mainly interested in the 3-dimensional case ... 1answer 405 views ### Operator theory of the Hessian How can I learn more about the operator theory of the Hessian? The Hessian of a function$u : \Omega \rightarrow \mathbb R$over a domain$\Omega \subseteq \mathbb R^n$is the matrix of second ... 0answers 162 views ### Intuition behind Stokes operator? I know that the definition of the Stokes operator (which appears in the functional form of the Navier-Stokes equations) is $$A = -P_L Δ$$ where$Δ$is the Laplacian, and$P_L$is the Leray ... 0answers 39 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$m$: ... 1answer 226 views ### A Poincare-Type Inequality and its generalization Let$f(\theta)$be a fixed positive$2\pi-$periodic$C^1$function on$\mathbb{R}$with $$\int_0^{2\pi}f(\theta)\cos\theta d\theta=\int_0^{2\pi}f(\theta)\sin\theta d\theta=0,$$ Does for any$2\pi-$... 2answers 780 views ### Are there any natural differential operators besides$d$? Let$\lambda = (\lambda_1, \ldots, \lambda_r)$and$\mu = (\mu_1, \ldots, \mu_r)$be partitions such that$\mu_j = \lambda_j +1$for one index$j$and$\mu_i = \lambda_i$for all other$i$. Then there ... 2answers 154 views ### Motivating the Bessel translation operator In a paper I am reading on the Hankel transform (this paper to be exact), I've come across a somewhat peculiar definition for a generalized translation operator. The operator is designed with a ... 2answers 233 views ### Diffusion on a semi-Riemannian manifold? A great deal of literature exists on the heat equation and heat kernel for a Riemannian manifold. The Laplace-Beltrami operator in the given metric replaces the flat Laplacian in the heat equation, ... 0answers 216 views ### Estimates of eigenvalues of elliptic operators on compact manifolds The classical Weyl law says that if$\Delta$is the Laplace operator on functions on a compact Riemannian manifold$(M^n,g)$,$n>2$, then its$k$th eigenvalue satisfies the asymptotic formula $$\... 0answers 171 views ### Discrete p-Laplacian One of the definitions of the discrete (weighted) p-Laplacian is the following:$$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$Consider the one dimensional case. Then the free ... 1answer 210 views ### adjoint of this closed (?) operator I am currently dealing with an unbounded operator T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow L^2(... 1answer 123 views ### Proper domain for operators in this paper on arxiv in equation 27, two operators$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$and$$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + \frac{mx}{\sqrt{1-x^2}... 1answer 156 views ### Functional Calculus and Fredholm index Let$-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that$-\Delta $is self-adjoint. Now, I am looking for closed operators$T$and$T^*of order ... 2answers 1k views ### Exact Definition of Dirac Operator Many definitions of the Dirac operator in the tradition of the Physics literature are hard to grasp for a mathematician. I would like to ask for a precise, general, definition of the Dirac operator ... 1answer 218 views ### Cauchy problem for an overdetermined system of PDE This question is strictly related to this one. Let us consider the differential system with constant coefficients $$\left(\begin{array}{ccc} B_{11} & B_{12} & 0\\ ... 1answer 129 views ### Heat Kernel estimate at the level of the form Let (M,g) be a compact Riemannian manifold. It is known there exist Gaussian estimates of the heat kernel and its derivatives acting on functions on M. The kind of estimate I'm looking for could ... 1answer 185 views ### How to solve this differential equation with an infinite sum? I would like to find solutions of the following differential equation: \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x) For example in space of function from \mathbb R^* to \mathbb ... 1answer 102 views ### Local fractional Sobolev inequality If \mathcal{X} is a smooth cutoff near 0 in \mathbb{R}^n, then M_0 = \mathcal{X}(-\Delta+Id)\mathcal{X} is a self-adjoint operator in L^2(\mathbb{R}^n). Because M_0 is semi-positive and the ... 4answers 1k views ### Green's operator of elliptic differential operator Let P:\Gamma(E)\rightarrow\Gamma(F) be an elliptic partial differential operator, with index =0 and closed image of codimension =1, between spaces \Gamma(E) and \Gamma(F) of smooth sections ... 0answers 104 views ### “simulteneous eigenvectors” under the full set of weighted Laplacians on a g-fold product of the Poincare half plane This question is closely related to the following MO question Characterizing the real analytic Eisenstein series Let \mathfrak{h}=\{z=x+iy\in\mathbf{C}\} be the Poincare upper half plane endowed ... 1answer 562 views ### Killing vector fields on sphere Let u be a smooth function on \mathbb S^2, and assume that for every killing vector field V on \mathbb S^2.$$\int_{\mathbb S^2} V(u) x_j dS=0\text{,}\forall j=1,2,3$$Is u necessarily ... 1answer 214 views ### When are the Dolbeault and de Rham dgas homotopy equivalent? Let M be a compact Kahler manifold. Then the Hodge decomposition says that the Dolbeault dga (of forms of all bidegree) and the de Rham dga on \Omega_{\mathbb C}^\bullet(M) have isomorphic ... 0answers 152 views ### Existence of solution? I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here. Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf Let \mathcal{... 0answers 88 views ### Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE A polynomial vector field of degree n on S^{2} is the Poincare compactification of a n degree polynomial vector field on \mathbb{R}^{2}.It is a real analytic vector field on S^{2} which ... 1answer 196 views ### Horizontal lift of differential operator On a Riemannian manifold M, there is a canonical horizontal lift X^{\mathrm{hor}} of vector fields X to TM, which is characterized by the two properties that X^{\mathrm{hor}} is a ... 0answers 124 views ### Elliptic PDE-Fredholm PDE(Is there a contradictory situation) Let E be a smooth vector bundle on a closed manifold M. Assume that D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E) is a diff. operator which is a fredhoolm operator, in the algebraic ... 0answers 147 views ### Is Laplacian a surjective operator? For a closed manifold the laplacian is almost surjective operator since the index of \Delta is zero and there is no a non constant harmonic function. So the codimension of the image ... 0answers 269 views ### The “Rolle theorem” for sections of a vector bundle 1) Assume that E\to M is a smooth real vector bundle and \nabla is a connection. (We do not assume any metric compatibility since we do not fix a metric on E). Assume that ... 2answers 601 views ### Elliptic operators corresponds to non vanishing vector fields Let X be a non vanishing vector field on a compact manifold M. The only differential operator associated with X which I am aware of, is the derivational operator D(g)=X.g. Unfortunately ... 0answers 265 views ### Noncommutative geometry and line length I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If D is the Dirac operator, he sets ds ... 1answer 168 views ### Euclidean Algorithm for differential operators While perusing through the article "Algorithms for solving linear ordinary differential equations" by Winfried Fakler (a pdf can be found through a google search), I noticed Faker mentioning on page 2 ... 1answer 80 views ### Pullback via flow as operator group Let X be a vector field on a manifold M that induces a complete flow \Theta_t. Then the operator family \Theta_t^*,$$\Theta_t^*u(x) = u(\Theta_t(x))is a strongly continuous semigroup of ... 0answers 31 views ### Differential operator with codimension 2 singularity in the domain The soft version of the question is as follows: suppose I have a linear operator, and I know it is a 'nice' differential operator on its domain minus a singular set of codimension two. Does the ... 1answer 490 views ### Helmholtz equation Poynting vector integral The Maxwell's equation for harmonic time dependent field in vacuum is \begin{align} \nabla \times B + i\omega E &= 0\\ \nabla \times E - i\omega B &= 0 \\ \nabla \cdot B &= 0 \\ \nabla \... 0answers 148 views ### Existence of the Dirichlet heat kernel for arbitrary open subsets? consider first of all an open and bounded subset \Omega\subset\mathbb{R}^n, s.t. the boundary \partial \Omega is a manifold of class C^2. Then I know that there exists a Dirichlet heat kernel, i.... 1answer 318 views ### Vector Laplace Beltrami operator of the Gauss map Consider an abstract surface (M,g) embedded into \mathbb{R}^3 via f:M \to \mathbb{R}^3. Denote by N:M \to \mathbb{R}^3 the Gauss map (normal field) of the surface. Write the Laplace Beltrami ... 1answer 340 views ### Yang-Mills equations are not elliptic [closed] How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic? Alternatively, how does one calculate the principal symbol of the Yang-Mills equations? Can ... 1answer 371 views ### Derivatives of infinite order [closed] Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature? For example, can one make sense of\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 \... 1answer 153 views ### Differential Operator Simplification Does anyone know the explicit formulation for theq_k$'s in, $$(x+D)^n=\sum_{k=0}^n q_k(x)D^k\ \ \ \ ?$$ I know that$e^{-x^2/2+x}$is a fixed point of$(x+D)$. I also, know that $$(x+D)H_n(x)e^{-x^... 0answers 186 views ### The Moyal action of a planar vector field Let X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y} be a polynomial vector field on \mathbb{R}^{2}. Consider the following (Moyal) operator on \mathbb{C}[x,y]: \tilde{D}_{X}(f)=... 1answer 127 views ### Laplacian on space of measures Let X be a compact Riemannian manifold and let \mathcal{M}(X) be the space of regular finite Borel measures with the total variation as norm. The Laplace-Belrami-Operator \Delta on X with ... 1answer 114 views ### Eigenvalue problem of an operator involving the exterior derivative of differential forms Consider two functions \alpha,\beta: \mathbb{R}^2 \to \mathbb{R}, where \alpha is given and we look for solutions \beta such that$$*(d\alpha \wedge d\beta) = \lambda \beta$$for some$\lambda ...
Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The ...