De Monvel Calculus Research Articles

Index theory for boundary value problems via continuous fields of [formula omitted]-algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid T − X generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field C r ∗ ( T − X ) of C ∗ -algebras over [ 0 , 1 ] . Its fiber in ℏ = 0 , C r ∗ ( T − X ) , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ℏ ≠ 0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K 0 ( C r ∗ ( T − X ) ) = K 0 ( C 0 ( T ∗ X ) ) → K 0 ( K ) = Z . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.

Elliptic boundary problems on manifolds with polycylindrical ends

We investigate general Shapiro–Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.

A K-theoretic proof of Boutet de Monvel's index theorem for boundary value problems

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory 0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

C* -structure and K-theory of Boutet de Monvels algebra

We consider the norm closure $A$ of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold $X$ with boundary $Y$. We first describe the image and the kernel of the continuous extension of the boundary principal symbol to $A$. If the $X$ is connected and $Y$ is not empty, we then show that the K-groups of $A$ are topologically determined. In case the manifold, its boundary and the tangent space of the interior have torsion-free K-theory, we prove that $K_i(A/K)$ is isomorphic to the direct sum of $K_i(C(X))$ and $K_{1-i}(C_0(TX'))$, for i=0,1, with $K$ denoting the compact ideal and $TX'$ the tangent bundle of the interior of $X$. Using Boutet de Monvel's index theorem, we also prove this result for i=1 without assuming the torsion-free hypothesis. We also give a composition sequence for $A$.

Parametrix construction for a class of anisotropic operators

We set Boutet de Monvel's Calculus for hypoelliptic operators (in the case of flat symplectic characteristic manifold) in a Weyl-Hormander framework that also contains anisotropically vanishing symbols. In this context we construct a parametrix for the related operators.

Dixmier's trace for boundary value problems

Let X be a smooth manifold with boundary of dimension n > 1. The operators of order −n and type zero in Boutet de Monvel's calculus form a subset of Dixmier's trace ideal \(\) for the Hilbert space \(\) of L2 sections in vector bundles E over X, F over ∂X.

The Noncommutative Residue for Manifolds with Boundary

We construct a trace on the algebra of classical elements in Boutet de Monvel's calculus on a compact manifold with boundary of dimensionn>2. This trace coincides with Wodzicki's noncommutative residue if the boundary is reduced to the empty set. Moreover, we show that it is the unique continuous trace on this algebra up to multiplication by a constant.

A characterization of the singular green operators in boutet de monvel's calculus via wedge sobolev spaces

The singular Green operators in Boutet de Monvel's calculus are characterized by the behavior of their iterated commutators with differentiations and vector fields tangential to the boundary on wedge Sobolev spaces.