Refine
Year of publication
Document Type
- Preprint (88)
- Article (40)
- Monograph/Edited Volume (27)
- Review (1)
Language
- English (156)
Keywords
- index (9)
- manifolds with singularities (6)
- Fredholm property (5)
- Toeplitz operators (5)
- Cauchy problem (4)
- Hodge theory (4)
- Navier-Stokes equations (4)
- pseudodifferential operators (4)
- star product (4)
- 'eta' invariant (3)
- Dirac operator (3)
- Dirichlet to Neumann operator (3)
- Neumann problem (3)
- Quasilinear equations (3)
- Riemann-Hilbert problem (3)
- boundary value problems (3)
- classical solution (3)
- differential operators (3)
- elliptic complexes (3)
- Beltrami equation (2)
- Cauchy data spaces (2)
- Clifford algebra (2)
- Euler-Lagrange equations (2)
- Fredholm operators (2)
- Heat equation (2)
- Laplace-Beltrami operator (2)
- Lefschetz number (2)
- Nonlinear Laplace operator (2)
- Removable sets (2)
- cohomology (2)
- elliptic operators (2)
- integral formulas (2)
- monodromy matrix (2)
- pseudodifferential operator (2)
- root functions (2)
- singular perturbation (2)
- trace (2)
- Analytic continuation (1)
- Analytic extension (1)
- Angular derivatives (1)
- Boundary value problem (1)
- Boundary value problems for first order systems (1)
- Capture into resonance (1)
- Carleman formulas (1)
- Composition operators (1)
- De Rham complex (1)
- Differential invariant (1)
- Dirichlet problem (1)
- Discontinuous Robin condition (1)
- Electromagnetic waves (1)
- Elliptic complex (1)
- Elliptic complexes (1)
- Equivalence (1)
- Euler equations (1)
- Extremal problem (1)
- Fischer-Riesz equations (1)
- Fredholm operator (1)
- G-index (1)
- G-trace (1)
- Green formula (1)
- Green formulas (1)
- Holomorphic map (1)
- Holomorphic mappings (1)
- Lagrangian system (1)
- Lamé system (1)
- Lipschitz domain (1)
- Lipschitz domains (1)
- Newton method (1)
- Non-coercive problem (1)
- Non-linear semigroups (1)
- Perturbed complexes (1)
- Porous medium equation (1)
- Primary: 47B35 (1)
- Pseudodifferential operators (1)
- Quasiconformal mapping (1)
- Root function (1)
- Scattering (1)
- Second order elliptic equations (1)
- Secondary: 47L80 (1)
- Semigroup (1)
- Stratton-Chu formulas (1)
- Sturm-Liouville problem (1)
- Sturm-Liouville problems (1)
- Symplectic manifold (1)
- Unit disk (1)
- WKB method (1)
- Zeta-function (1)
- analytic continuation (1)
- asymptotic expansion (1)
- asymptotic methods (1)
- asymptotics (1)
- boundary layer (1)
- boundary value problem (1)
- boundary values problems (1)
- characteristic boundary point (1)
- characteristic points (1)
- composition operator (1)
- curvature (1)
- cusp (1)
- dbar-Neumann problem (1)
- de Rham complex (1)
- discontinuous Robin condition (1)
- division of spaces (1)
- divisors (1)
- domains with singularities (1)
- dynamical system (1)
- elliptic complex (1)
- ellipticity with parameter (1)
- equivalence (1)
- evolution equation (1)
- first boundary value problem (1)
- fundamental solution (1)
- geometric optics approximation (1)
- heat equation (1)
- inegral formulas (1)
- integral representation method (1)
- invariant (1)
- inverse problems (1)
- lattice packing and covering (1)
- lattice point (1)
- logarithmic residue (1)
- manifold with boundary (1)
- manifolds with cusps (1)
- manifolds with edges (1)
- mapping degree (1)
- matching of asymptotic expansions (1)
- meromorphic family (1)
- mixed problems (1)
- non-coercive boundary conditions (1)
- nonlinear equations (1)
- nonlinear semigroup (1)
- nonsmooth curves (1)
- p-Laplace Operator (1)
- p-Laplace equation (1)
- p-Laplace operator (1)
- polyhedra and polytopes (1)
- porous medium equation (1)
- quasiconformal mapping (1)
- quasilinear Fredholm operator (1)
- quasilinear equation (1)
- regular figures (1)
- regularisation (1)
- regularization (1)
- relative cohomology (1)
- removable set (1)
- removable sets (1)
- residue (1)
- singular integral equations (1)
- singular point (1)
- small parameter (1)
- spectral kernel function (1)
- spectral theorem (1)
- spirallike function (1)
- strongly pseudoconvex domains (1)
- symmetry group (1)
- symplectic manifold (1)
- the Dirichlet problem (1)
- the first boundary value problem (1)
- weak boundary values (1)
- weighted Hölder spaces (1)
- weighted spaces (1)
- ∂-operator (1)
We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer.
We define the Dirichlet to Neumann operator for an elliptic complex of first order differential operators on a compact Riemannian manifold with boundary. Under reasonable conditions the Betti numbers of the complex prove to be completely determined by the Dirichlet to Neumann operator on the boundary.
For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.
The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.
For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.
We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.
When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for
the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on
a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two.
We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.
We consider a solution of the nonlinear Klein-Gordon equation perturbed by a parametric driver. The frequency of parametric perturbation varies slowly and passes through a resonant value, which leads to a solution change. We obtain a new connection formula for the asymptotic solution before and after the resonance.
Given a system of entire functions in Cn with at most countable set of common zeros, we introduce the concept of zeta-function associated with the system. Under reasonable assumptions on the system, the zeta-function is well defined for all s ∈ Zn with sufficiently large components. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in Cn.