85 research outputs found
Bounded Imaginary Powers of Differential Operators on Manifolds with Conical Singularities
We study the minimal and maximal closed extension of a differential operator
A on a manifold B with conical singularities, when A acts as an unbounded
operator on weighted L^p-spaces over B, 1 < p < \infty. Under suitable
ellipticity assumptions we can define a family of complex powers A^z. We also
obtain sufficient information on the resolvent of A to show the boundedness of
the purely imaginary powers. Examples concern unique solvability and maximal
regularity for the solution of the Cauchy problem for the Laplacian on conical
manifolds as well as certain quasilinear diffusion equations.Comment: 27 pages, 3 figures (revised version 23/04/'02
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.Comment: Title slightly changed. Accepted for publication in Journal fuer die
reine und angewandte Mathemati
C*-Structure and K-Theory of Boutet de Monvel's Algebra
We consider the norm closure of the algebra of all operators of order and
class zero in Boutet de Monvel's calculus on a manifold with boundary .
We first describe the image and the kernel of the continuous extension of the
boundary principal symbol to . If the is connected and is not empty,
we then show that the K-groups of are topologically determined. In case the
manifold, its boundary and the tangent space of the interior have torsion-free
K-theory, we prove that is isomorphic to the direct sum of
and , for i=0,1, with denoting the compact
ideal and the tangent bundle of the interior of . 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 .Comment: Final version, to appear in J. Reine Angew. Math. Improved
K-theoretic result
Families index for Boutet de Monvel operators
We define the analytical and the topological indices for continuous families of operators in the C*-closure of the Boutet de Monvel algebra. Using techniques of C*-algebra, K-theory, and the Atiyah–Singer theorem for families of elliptic operators on a closed manifold, we prove that these two indices coincide
Bounded -Calculus for Differential Operators on Conic Manifolds with Boundary
We derive conditions that ensure the existence of a bounded
-calculus in weighted -Sobolev spaces for closed extensions
of a differential operator on a conic manifold with
boundary, subject to differential boundary conditions . In general, these
conditions ask for a particular pseudodifferential structure of the resolvent
in a sector . In
case of the minimal extension they reduce to parameter-ellipticity of the
boundary value problem . Examples concern the Dirichlet and Neumann
Laplacians.Comment: 23 page
On the Fredholm property of bisingular pseudodifferential operators
For operators belonging either to a class of global bisingular
pseudodifferential operators on or to a class of bisingular
pseudodifferential operators on a product of two closed smooth
manifolds, we show the equivalence of their ellipticity (defined by the
invertibility of certain associated homogeneous principal symbols) and their
Fredholm mapping property in associated scales of Sobolev spaces. We also prove
the spectral invariance of these operator classes and then extend these results
to the even larger classes of Toeplitz type operators.Comment: 21 pages. Expanded sections 3 and 4. Corrected typos. Added
reference
Realizations of Differential Operators on Conic Manifolds with Boundary
We study the closed extensions (realizations) of differential operators
subject to homogeneous boundary conditions on weighted L_p-Sobolev spaces over
a manifold with boundary and conical singularities. Under natural ellipticity
conditions we determine the domains of the minimal and the maximal extension.
We show that both are Fredholm operators and give a formula for the relative
index.Comment: 41 pages, 1 figur
Adaptive FE-BE Coupling for Strongly Nonlinear Transmission Problems with Coulomb Friction
We analyze an adaptive finite element/boundary element procedure for scalar
elastoplastic interface problems involving friction, where a nonlinear
uniformly monotone operator such as the p-Laplacian is coupled to the linear
Laplace equation on the exterior domain. The problem is reduced to a
boundary/domain variational inequality, a discretized saddle point formulation
of which is then solved using the Uzawa algorithm and adaptive mesh refinements
based on a gradient recovery scheme. The Galerkin approximations are shown to
converge to the unique solution of the variational problem in a suitable
product of L^p- and L^2-Sobolev spaces.Comment: 27 pages, 3 figure
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