Related papers: On differential operators generated by geometric s…
We provide a general framework to study invariant properties of various gradient-like and Laplace-like differential operators naturally associated to geometric structures on $\mathbb{R}^n$, which encompass Euclidean, Minkowski,…
We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary…
We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential…
We characterize all natural linear operations between spaces of differential forms on contact manifolds. Our main theorem says roughly that such operations are built from some algebraic operators which we introduce and the exterior…
We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived…
The problem of equivalency for linear differential operators of the first order is discussed.
In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of…
The analysis of mathematical structure of the method of operator manifold guides our discussion. The latter is a still wider generalization of the method of secondary quantization with appropriate expansion over the geometric objects. The…
We prove the boundedness on $L^p$, $1<p<\infty$, of operators on manifolds which arise by taking conditional expectation of transformations of stochastic integrals. These operators include various classical operators such as second order…
We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from…
This thesis generalizes the differential operators on standard oriented graphs and oriented hypergraphs introduced in 10.1137/15M1022793 and arXiv:2007.00325. The extended concepts of gradients, adjoints and $p$-Laplacians for vertices and…
We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of…
In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this…
This note describes the construction of c U p-invariant differential operators on statistical manifolds, i.e. of operators canonically associated to a geometry which synthetizes the properties of conformal and projective geometries.
We investigate spectral functionals associated with Dirac and Laplace-type differential operators on manifolds, defined via the Wodzicki residue, extending classical results for Dirac operators derived from the Levi-Civita connection to…
We consider generalized gradients in the general context of $G$-structures. They are natural first order differential operators acting on sections of vector bundles associated to irreducible $G$-representations. We study their geometric…
A mechanism deriving new well-posed evolutionary equations from given ones is inspected. It turns out that there is one particular spatial operator from which many of the standard evolutionary problems of mathematical physics can be…
The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of…
We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each…
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation…