Refine
Year of publication
Document Type
- Monograph/Edited Volume (121)
- Preprint (114)
- Article (100)
- Conference Proceeding (1)
- Postprint (1)
Keywords
- Fredholm property (6)
- boundary value problems (6)
- elliptic operators (6)
- manifolds with singularities (6)
- index (5)
- pseudodifferential operators (5)
- Boundary value problems (4)
- Pseudo-differential operators (4)
- ellipticity (4)
- relative index (4)
- 'eta' invariant (3)
- Atiyah-Bott condition (3)
- Atiyah-Bott obstruction (3)
- K-theory (3)
- Mellin transform (3)
- Zaremba problem (3)
- conical singularities (3)
- conormal symbol (3)
- differential operators (3)
- index theory (3)
- pseudo-differential boundary value problems (3)
- Asymptotics of solutions (2)
- Atiyah-Patodi-Singer theory (2)
- Edge calculus (2)
- Fredholm operators (2)
- Lefschetz fixed point formula (2)
- Meromorphic operator-valued symbols (2)
- corner Sobolev spaces with double weights (2)
- edge singularities (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic complexes (2)
- elliptic families (2)
- elliptic family (2)
- elliptic operator (2)
- homotopy classification (2)
- index formulas (2)
- manifold with singularities (2)
- manifolds with conical singularities (2)
- manifolds with edges (2)
- monodromy matrix (2)
- operator-valued symbols (2)
- pseudo-differential operators (2)
- pseudodiferential operators (2)
- quantization (2)
- regularizer (2)
- regularizers (2)
- spectral flow (2)
- surgery (2)
- symmetry conditions (2)
- weighted edge spaces (2)
- 35J70 (1)
- 47G30 (1)
- 58J40 (1)
- APS problem (1)
- Anisotropic pseudo-differential operators (1)
- Atiyah-Singer theorem (1)
- Boundary-contact problems (1)
- C0−semigroup (1)
- Calderón projections (1)
- Casped plates (1)
- Categories of stratified spaces (1)
- Cauchy Riemann operator (1)
- Chern character (1)
- Cone and edge pseudo-differential operators (1)
- Corner boundary value problems (1)
- Corner pseudo-differential operators (1)
- Crack theory (1)
- Edge degenerate operators (1)
- Edge symbols (1)
- Edge-degenerate operators (1)
- Elliptic complexes (1)
- Elliptic operators in domains with edges (1)
- Ellipticity and parametrices (1)
- Ellipticity of corner-degenerate operators (1)
- Ellipticity of edge-degenerate operators (1)
- Euler operator (1)
- Fourier and Mellin transform (1)
- Fourier and Mellin transforms (1)
- Fourier transform (1)
- G-index (1)
- G-trace (1)
- Green and Mellin edge operators (1)
- Hardy‘s inequality (1)
- Hodge theory (1)
- Korn’s weighted inequality (1)
- Lefschetz number (1)
- Manifolds with boundary (1)
- Mellin (1)
- Mellin and Green operators edge symbols (1)
- Mellin operators (1)
- Mellin oscillatory integrals (1)
- Mellin quantization (1)
- Mellin quantizations (1)
- Mellin symbols with values in the edge calculus (1)
- Meromorphic operator functions (1)
- Operator-valued symbols (1)
- Operator-valued symbols of Mellin type (1)
- Operators on manifolds with conical singularities (1)
- Operators on manifolds with edge (1)
- Operators on manifolds with edge and conical exit to infinity (1)
- Operators on manifolds with second order singularities (1)
- Operators on singular cones (1)
- Operators on singular manifolds (1)
- Pseudo-differential algebras (1)
- Pseudodifferential operators (1)
- Quantizations (1)
- Schrodinger equation (1)
- Singular analysis (1)
- Sobolev spaces with double weights on singular cones (1)
- Stratified spaces (1)
- Surface potentials with asymptotics (1)
- Toeplitz operators (1)
- Toeplitz-type pseudodifferential operators (1)
- Twisted symbolic estimates (1)
- Volterra operator (1)
- Volterra symbols (1)
- Weighted edge spaces (1)
- absorbing set (1)
- algebra (1)
- analytic index (1)
- asymptotic properties of eigenfunctions (1)
- asymptotics of solutions (1)
- boundary values problems (1)
- cohomology (1)
- conormal asymptotics (1)
- conormal symbols (1)
- contact transformations (1)
- continuity in Sobolev spaces with double weights (1)
- corner parametrices (1)
- degenerate elliptic systems (1)
- distribution with asymptotics (1)
- divisors (1)
- edge Sobolev spaces (1)
- edge algebra (1)
- edge quantizations (1)
- edge spaces (1)
- edge symbol (1)
- elliptic operators in subspaces (1)
- elliptic operators on non-compact manifolds (1)
- elliptic problem (1)
- ellipticity in the edge calculus (1)
- ellipticity of cone operators (1)
- ellipticity of corners operators (1)
- ellipticity with interface conditions (1)
- ellipticity with respect to interior and edge symbols (1)
- eta-invariant (1)
- exit calculus (1)
- exponential stability (1)
- exterior tensor product (1)
- index formula (1)
- index of elliptic operator (1)
- index of elliptic operators in subspaces (1)
- integral formulas (1)
- interfaces with conical singularities (1)
- iterated asymptotics (1)
- manifold with edge (1)
- manifolds with corners (1)
- manifolds with cusps (1)
- manifolds with edge and boundary (1)
- many-electron systems (1)
- meromorphic family (1)
- mixed elliptic problems (1)
- mod k index (1)
- nonhomogeneous boundary value problems (1)
- norm estimates with respect to a parameter (1)
- operator algebras on manifolds with singularities (1)
- operator calculus (1)
- operator valued symbols (1)
- operators on manifolds with conical and edge singularities (1)
- operators on manifolds with edges (1)
- operators on manifolds with singularities (1)
- operators with corner symbols (1)
- order reduction (1)
- parameter-dependent cone operators (1)
- parameter-dependent ellipticity (1)
- parameter-dependent pseudodifferential operators (1)
- parametrices of elliptic operators (1)
- principal symbolic hierarchies (1)
- problem of classification (1)
- pseudo-diferential operators (1)
- pseudo-differentialboundary value problems (1)
- pseudodifferential boundary value problems (1)
- pseudodifferential operator (1)
- pseudodifferential subspaces (1)
- relative cohomology (1)
- relative index formulas (1)
- residue (1)
- semiprocess (1)
- singular manifolds (1)
- spectral boundary value problems (1)
- spectral resolution (1)
- star product (1)
- symbols (1)
- symmetry group (1)
- symplectic (canonical) transformations (1)
- uniform compact attractor (1)
- vibration (1)
- weighted Sobolev spaces (1)
- weighted edge and corner spaces (1)
- weighted spaces (1)
- weighted spaces with asymptotics (1)
- ∂-operator (1)
The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.
We establish essential steps of an iterative approach to operator algebras, ellipticity and Fredholm property on stratified spaces with singularities of second order. We cover, in particular, corner-degenerate differential operators. Our constructions are focused on the case where no additional conditions of trace and potential type are posed, but this case works well and will be considered in a forthcoming paper as a conclusion of the present calculus.
We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.
We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro-Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces.
Operators on manifolds with corners that have base configurations with geometric singularities can be analysed in the frame of a conormal symbolic structure which is in spirit similar to the one for conical singularities of Kondrat'ev's work. Solvability of elliptic equations and asymptotics of solutions are determined by meromorphic conormal symbols. We study the case when the base has edge singularities which is a natural assumption in a number of applications. There are new phenomena, caused by a specific kind of higher degeneracy of the underlying symbols. We introduce an algebra of meromorphic edge operators that depend on complex parameters and investigate meromorphic inverses in the parameter-dependent elliptic case. Among the examples are resolvents of elliptic differential operators on manifolds with edges.
We study operators on singular manifolds, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. We introduce asymptotic parametrices, using tools from cone and edge pseudo-differential algebras. Our structures are motivated by models of many-particle physics with singular Coulomb potentials that contribute higher order singularities in Euclidean space, determined by the number of particles.
We study the asymptotic regularity of solutions of Hartree-Fock equations for Coulomb systems. In order to deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo-differential operators on conical manifolds. First, the non-self-consistent-field case is considered which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator which guarantee ellipticity for the local part of the Fock operator on the open stretched cone R+ × S². This proves existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed-point approach based on Cances and Le Bris analysis of the level-shifting algorithm, we show via another bootstrap argument, that the corresponding self-consistent-field solutions of the Hartree-Fock equation have the same type of asymptotic regularity.
We study pseudodifferential operators on a cylinder IR x B with cross section B that conical singularities. Configurations of that kind are the local model of cornere singularities with base spaces B. Operators A in our calculus are assumed to have symbols α which are meromorphic in the complex covariable with values in the space of all cone operators on B. In case α is dependent of the axial variable t ∈ IR, we show an explicit formula for solutions of the homogeneous equation. Each non-bjectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula.
Boundary value problems on manifolds with conical singularities or edges contain potential operators as well as trace and Green operators which play a similar role as the corresponding operators in (pseudo-differential) boundary value problems on a smooth manifold. There is then a specific asymptotic behaviour of these operators close to the singularities. We characterise potential operators in terms of actions of cone or edge pseudo-differential operators (in the neighbouring space) on densities supported by sbmanifolds which also have conical or edge singularities. As a byproduct we show the continuity of such potentials as continuous perators between cone or edge Sobolev spaces and subspaces with asymptotics.
We establish a calculus of boundary value problems (BVPs) on a manifold N with boundary and edge, based on Boutet de Monvel’s theory of BVPs in the case of a smooth boundary and on the edge calculus, where in the present case the model cone has a base which is a compact manifold with boundary. The corresponding calculus with boundary and edge is a unification of both structures and controls different operator-valued symbolic structures, in order to obtain ellipticity and parametrices.
We show relative index formulas for boundary value problems in cylindrical domains and Sobolev spaces with different weights at too. The amplitude functions are meromorphic in the axial covariable and take values in the space of boundary value problems on the cross section of the cylinder. Copyright (c) 2005 John Wiley & Sons, Ltd
Boundary value problems in Boutet de Monvel's algebra for manifolds with conical singularities II
(1995)
Boundary value problems in Boutet de Monvelïs algebra for manifolds with conical singularities I
(1994)
The paper is devoted to pseudodifferential boundary value problems in domains with cuspidal wedges. Concerning the geometry we even admit a more general behaviour, namely oscillating cuspidal wedges. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to edges.
Edge representations of operators on closed manifolds are known to induce large classes of operators that are elliptic on specific manifolds with edges, cf. [9]. We apply this idea to the case of boundary value problems. We establish a correspondence between standard ellipticity and ellipticity with respect to the principal symbolic hierarchy of the edge algebra of boundary value problems, where an embedded submanifold on the boundary plays the role of an edge. We first consider the case that the weight is equal to the smoothness and calculate the dimensions of kernels and cokernels of the associated principal edge symbols. Then we pass to elliptic edge operators for arbitrary weights and construct the additional edge conditions by applying relative index results for conormal symbols.
We study elliptic boundary value problems in a wedge with additional edge conditions of trace and potential type. We compute the (difference of the) number of such conditions in terms of the Fredholm index of the principal edge symbol. The task will be reduced to the case of special opening angles, together with a homotopy argument.
We construct a new calculus of boundary value problems with the transmission property on a non-compact smooth manifold with boundary and conical exits to infinity. The symbols are classical both in covariables and variables. The operators are determined by principal symbol tuples modulo operators of lower orders and weights (such remainders are compact in weighted Sobolev spaces). We develop the concept of ellipticity, construct parametrices within the algebra and obtain the Fredholm property. For the existence of Shapiro-Lopatinskij elliptic boundary conditions to a given elliptic operator we prove an analogue of the Atiyah-Bott condition.