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Keywords
- elliptic operators (6)
- manifolds with singularities (6)
- Fredholm property (5)
- boundary value problems (5)
- index (5)
- pseudodifferential operators (5)
- Boundary value problems (4)
- relative index (4)
- 'eta' invariant (3)
- Atiyah-Bott condition (3)
- K-theory (3)
- conical singularities (3)
- conormal symbol (3)
- differential operators (3)
- ellipticity (3)
- index theory (3)
- Atiyah-Bott obstruction (2)
- Atiyah-Patodi-Singer theory (2)
- Fredholm operators (2)
- Lefschetz fixed point formula (2)
- Mellin transform (2)
- Zaremba problem (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)
- pseudo-differential boundary value problems (2)
- pseudodiferential operators (2)
- quantization (2)
- regularizer (2)
- regularizers (2)
- spectral flow (2)
- surgery (2)
- symmetry conditions (2)
- weighted edge spaces (2)
- APS problem (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)
- Corner boundary value problems (1)
- Crack theory (1)
- Edge-degenerate operators (1)
- Elliptic operators in domains with edges (1)
- Euler operator (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)
- Meromorphic operator functions (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)
- Pseudo-differential operators (1)
- Pseudodifferential operators (1)
- Sobolev spaces with double weights on singular cones (1)
- Surface potentials with asymptotics (1)
- Toeplitz operators (1)
- Volterra symbols (1)
- absorbing set (1)
- analytic index (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 Sobolev spaces with double weights (1)
- degenerate elliptic systems (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)
- 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)
- manifold with edge (1)
- manifolds with cusps (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)
- operators on manifolds with conical and edge singularities (1)
- operators on manifolds with edges (1)
- operators on manifolds with singularities (1)
- order reduction (1)
- parameter-dependent cone operators (1)
- parameter-dependent ellipticity (1)
- parameter-dependent pseudodifferential operators (1)
- principal symbolic hierarchies (1)
- problem of classification (1)
- pseudo-diferential operators (1)
- pseudo-differential 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)
- spectral boundary value problems (1)
- spectral resolution (1)
- star product (1)
- symmetry group (1)
- symplectic (canonical) transformations (1)
- uniform compact attractor (1)
- vibration (1)
- weighted spaces (1)
- weighted spaces with asymptotics (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (114)
The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.
We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved.
The homotopy classification and the index of boundary value problems for general elliptic operators
(1999)
We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition.
We prove the existence of a limit in Hm(D) of iterations of a double layer potential constructed from the Hodge parametrix on a smooth compact manifold with boundary, X, and a crack S ⊂ ∂D, D being a domain in X. Using this result we obtain formulas for Sobolev solutions to the Cauchy problem in D with data on S, for an elliptic operator A of order m ≥ 1, whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of the double layer potential. A similar regularisation is constructed also for a mixed problem in D.