Cobordism invariance of the index of Callias-type operators

We introduce a notion of cobordism of Callias-type operators overcomplete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application, we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two noncompact but topologically simpler manifolds. As another application, we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.     

Singular functions and relative index for elliptic corner operators

We show an explicit link between the nature of a singular point and the behaviour of the coefficients of the equation, under which formal asymptotic expansions are still available. We also derive a general relative index theorem for elliptic operators. To the memory of Lamberto Cattabriga     

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.     

‘Twice’ Equivariant C*-Index Theorem and the Index Theorem for Families

We prove the index theorem for elliptic operators acting on sections of bundles where the fiber is equal to a projective module over a C ^*-algebra in the situation of the action of a compact Lie group on this algebra as well as on the total space commuting with a symbol. For this purpose, we prove in particular the corresponding Thom isomorphism theorem. As an application, the equivariant family index theorem for a direct product of a base by the space of parameters is obtained.     

An Index Theorem on Foliated Flat Bundles

In this paper we shall study the indices of some leafwise elliptic operators on foliated bundles, by using some concretely constructed cyclic cocycles on the smooth foliation algebras. The key is a decomposition of the KK-classes associated to the operators. The group action analogue of this approach provides a bridge from the classical Atiyah-Singer index theorem to the higher -index theorem of Connes and Moscovici.     

Index of nonlocal problems associated with a bundle

We study operators associated with a bundle with compact base and fiber. We construct an algebra of such operators. For elliptic elements of the algebra, we prove the finiteness theorem and derive an index formula.     

Elliptic Quasicomplexes on Compact Closed Manifolds

We consider quasicomplexes of pseudodifferential operators on a smooth compact manifold without boundary. To each quasicomplex we associate a complex of symbols. The quasicomplex is elliptic if this symbol complex is exact away from the zero section. We prove that elliptic quasicomplexes are Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes and prove a generalisation of the Atiyah–Singer index theorem.     

The index of geometric operators on Lie groupoids

We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the “topological side” of the index theorem. This results in index formulae for Lie groupoid analogues of the familiar geometric operators on manifolds such as the signature and Dirac operator expressed in terms of the usual characteristic classes in Lie algebroid cohomology.     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Index formulas for stratified manifolds

We consider elliptic operators on stratified manifolds with stratification of arbitrary length. Under some (symmetry-like) conditions imposed on the symbols of these operators, we obtain index formulas in which the index of an operator is expressed as the sum of indices of some (explicitly written out) elliptic operators on the strata.     

de Rham－Hodge－Signature算子的局部指标定理

整体的Ａｔｉｙａｈ－Ｓｉｎｇｅｒ指标定理对于一般的椭圆微分算子都成立．但对于局部指标定理来说，人们只能对具体的算子给予证明．并且各种情况需个别处理．由［６］知本文中证得的ｄｅＲｈａｍ－Ｈｏｄｇｅ－Ｓｉｇｎａｔｕｒｅ算子的局部指标既不是α型也不是β型的．这和经典的椭圆算子情形不同．     

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.     

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.     

Trace Theorems for Anisotropic Weighted SOBOLEV Spaces in a Corner

In this paper, we prove a trace theorem for anisotropic weighted Sobolev spaces in a cube Q naturally associated to a class of degenerate elliptic operators. The fundamental property of this class is the existence of a suitable metric d which is “natural” for the operators. The basic tool of the proof is a representation formula obtained via suitable non-euclidean translations closely fitting the geometry of the d-balls. In a more particular situation, we construct a right inverse of the trace operator and we describe the compatibility conditions on the edges of Q.     

Regularity Criteria for the Generalized MHD Equations

For Sobolev spaces in Lipschitz domains with no imposed boundary conditions, the Aronszajn–Smith theorem algebraically characterizes coercive formally positive integro-differential quadratic forms. Recently, linear elliptic differential operators with formally positive forms have been constructed with the property that no formally positive forms for these operators can be coercive in any bounded domain. In the present article 4th order operators of this kind are shown by perturbation to have coercive forms that are (necessarily) algebraically indefinite. The perturbation here from noncoercive formally positive forms to coercive algebraically indefinite forms requires Agmon's characterization of coerciveness in smoother domains than Lipschitz.     

K-theory of toeplitzC *-algebras on Lie spheres

In this paper we compute theK-groups of theC ^*-algebra of Toeplitz operators on the Lie spheres. As a corollary we get an index theorem for Toeplitz operators with matricial symbols analogous to the index theorem of Berger, Coburn and Koranyi for Toeplitz operators with scalar valued symbols.     

K-homology and index theory on contact manifolds

This paper applies K-homology to solve the index problem for a class of hypoelliptic (but not elliptic) operators on contact manifolds. K-homology is the dual theory to K-theory. We explicitly calculate the K-cycle (i.e., the element in geometric K-homology) determined by any hypoelliptic Fredholm operator in the Heisenberg calculus. The index theorem of this paper precisely indicates how the analytic versus geometric K-homology setting provides an effective framework for extending formulas of Atiyah–Singer type to non-elliptic Fredholm operators.     

Elliptic Operators in Subspaces and the Eta Invariant

In this paper we obtain a formula for the fractional part of the -invariant for elliptic self-adjoint operators in topological terms. The computation of the -invariant is based on the index theorem for elliptic operators in subspaces obtained by Savin and Sternin. We also apply the K-theory with coefficients _n . In particular, it is shown that the group K(T ^* M, _n ) is realized by elliptic operators (symbols) acting in appropriate subspaces.     

The family index theorem and bifurcation of solutions of nonlinear elliptic BVP

We obtain some new bifurcation criteria for solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. The results are of different nature from the ones that can be obtained via the traditional Lyapunov–Schmidt reduction. Our sufficient conditions for bifurcation are derived from the Atiyah–Singer family index theorem and therefore they depend only on the coefficients of derivatives of leading order of the linearized differential operators. They are computed explicitly from the coefficients without the need of solving the linearized equations. Moreover, they are stable under lower order perturbations.     

On non-linear elliptic equation with weight

In this paper, we prove the existence and uniqueness theorem of the Dirichlet problem for certain non-linear elliptic operators with Muckenhoupt weight.